节: 线性系统图图 | 7.1 线性系统图

章节大纲

Graphs of Linear Systems
::线性系统图图

Determine Whether an Ordered Pair is a Solution to a System of Equations
::确定有秩序对等是否是公式体系的解决方案

A linear system of equations is a set of equations that must be solved together to find the one solution that fits them both.
::线性方程式系统是一套方程式,必须一起解决,才能找到适合两者的解决办法。

Consider this system of equations :
::考虑一下这个方程系统:

$\begin{align*}\begin{cases}y = x + 2\\ y = -2x + 1 \end{cases}\end{align*}$
::{y=x+2y=%2x+1

Since the two lines are in a system, we deal with them together by graphing them on the same coordinate axes. We can use any method to graph them; let’s do it by making a table of values for each line.
::因为两条线都在一个系统中,所以我们通过在同一坐标轴上绘制图表来共同处理这两条线。我们可以使用任何方法来绘制两条线的图表;让我们通过为每条线绘制一个数值表来做到这一点。

Line 1: $\begin{align*}y=x+2\end{align*}$
::第1行:y=x+2

$\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 0 & 2 \\\hline 1 & 3 \\\hline \end{array}\end{align*}$
::xy0213 xy0213

Line 2: $\begin{align*}y=\text{-}2x=1\end{align*}$
::第2行 y= 2x= 1

$\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 0 & 1 \\\hline 1 & \text{-}1 \\\hline \end{array}\end{align*}$
::xy011-1

We already know that any point that lies on a line is a solution to the equation for that line. That means that any point that lies on both lines in a system is a solution to the whole system .
::我们已经知道,线上的任何点就是线的方程的解决方案。这意味着,系统中两条线上的任何点就是整个系统的解决办法。

So in this system:
::因此在这个系统中:

Point $\begin{align*}A\end{align*}$ is not a solution to the system because it does not lie on either of the lines.
::A点不是系统的解决办法,因为它不取决于任何一条线。

Point $\begin{align*}B\end{align*}$ is not a solution to the system because it lies only on the blue line but not on the red line.
::B点不是系统的解决办法,因为它只存在于蓝线上,而不是红线上。

Point $\begin{align*}C\end{align*}$ is a solution to the system because it lies on both lines at the same time.
::C点是系统的一个解决方案,因为它同时处于两条线上。

In fact, point $\begin{align*}C\end{align*}$ is the only solution to the system, because it is the only point that lies on both lines. For a system of equations, the geometrical solution is the intersection of the two lines in the system. The algebraic solution is the ordered pair that solves both equations - in other words, it is the coordinates of the intersection point.
::事实上,C点是这个系统的唯一解决方案,因为它是两个线唯一的解决方案。对于一个方程系统,几何解决方案是系统中两个线的交叉点。代数解决方案是解决两个方程的定单对 — — 换句话说,它就是交叉点的坐标。

You can confirm the solution by plugging it into the system of equations, then checking that the solution works in each equation.
::您可以通过将其插入方程式系统来确认解决方案, 然后检查解决方案在每个方程式中有效。

Finding the Solution to a System of Equations
::寻找公式体系的解决方案

Determine which of the points (1, 3), (0, 2), or (2, 7) is a solution to the following system of equations:
::确定哪些点(1、3、0、2)或(2、7)是下列方程系统的解决办法:

$\begin{align*}\begin{cases}y = 4x - 1\\ y = 2x + 3 \end{cases}\end{align*}$
::{y=4x-1y=2x+3 {y=4x-1y=2x+3

To check if a coordinate point is a solution to the system of equations, we plug each of the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ values into the equations to see if they work.
::要检查一个坐标点是否是公式体系的解决方案, 我们将在公式中插入每个 x 和 y 值, 看看它们是否有效。

Point (1, 3):
::点(1,3):

$\begin{align*}y &= 4x - 1\\ 3 & \overset{?}{=} 4(1) - 1\\ 3 &= 3 \ \text{solution checks}\\ \\ y &= 2x + 3\\ 3 &\overset{?}{=} 2(1) + 3\\ 3 &\neq 5 \ \text{solution does not check}\\\end{align*}$
::y=4x- 13? =4(1)- 13=3 溶液检查=2x+33?=2(1)+33=5 溶液不检查

Point (1, 3) is on the line $\begin{align*}y = 4x - 1\end{align*}$ , but it is not on the line $\begin{align*}y = 2x + 3\end{align*}$ , so it is not a solution to the system.
::点(1, 3)在 y= 4x-1 线上, 但不是在 y= 2x+3 线上, 所以不是系统的解决办法。

Point (0, 2):
::点( 0, 2) :

$\begin{align*}y &= 4x - 1\\ 2 &\overset{?}{=} 4(0) - 1\\ 2 &\neq -1 \ \text{solution does not check}\end{align*}$
::y=4x- 12?=4(0)- 12*1 解决方案不检查

Point (0, 2) is not on the line $\begin{align*}y = 4x - 1\end{align*}$ , so it is not a solution to the system. Note that it is not necessary to check the second equation because the point needs to be on both lines for it to be a solution to the system.
::点 (0, 2) 不在 y= 4x-1 线上, 因此它不是系统的解决办法。请注意, 没有必要检查第二个等式, 因为点必须在两条线上才能成为系统的解决办法。

Point (2, 7):
::点(2,7):

$\begin{align*}y &= 4x - 1\\ 7 &\overset{?}{=} 4(2) - 1\\ 7 &= 7 \ \text{solution checks}\\ \\ y &= 2x + 3\\ 7 &\overset{?}{=} 2(2) + 3\\ 7 &= 7 \ \text{solution checks}\end{align*}$
::y=4x- 17? =4(2)- 17=7 溶液检查 =2x+37? =2(2)+37=7 溶液检查

Point (2, 7) is a solution to the system since it lies on both lines.
::点(2、7)是系统的解决办法,因为它是两条线上的。

The solution to the system is the point (2, 7).
::系统的解决办法是点(2、7)。

Determine the Solution to a Linear System by Graphing
::通过图形绘制确定线性系统的解决方案

The solution to a linear system of equations is the point, (if there is one) that lies on both lines. In other words, the solution is the point where the two lines intersect.
::线性方程体系的解决方案是两个线条的点( 如果有的话 ) 。换句话说, 解决方案是两条线交错的点。

We can solve a system of equations by graphing the lines on the same coordinate plane and reading the intersection point from the graph.
::我们可以通过在同一坐标平面上绘制线条图和从图中读取交叉点来解决方程式系统。

This method most often offers only approximate solutions, so it’s not sufficient when you need an exact answer. However, graphing the system of equations can be a good way to get a sense of what’s really going on in the problem you’re trying to solve, especially when it’s a real-world problem.
::这种方法通常只提供近似的解决办法,所以当你需要准确的答案时是不够的。然而,图形化方程系统可以是一个很好的方法,使人们了解你试图解决的问题中到底发生了什么,特别是当这是一个现实世界的问题时。

Solving a System of Equations by Graphing
::通过图形绘制解决方平系统

Solve the system of equations by graphing:
::以图示方式解决方程式系统:

$\begin{align*}\begin{cases} y = 3x - 5\\ y = \text{-}2x + 5 \end{cases}\end{align*}$
::{y=3x-5y=-2x+5

Graph both lines on the same coordinate axis using any method you like.
::使用任何您喜欢的方法在同一坐标轴上绘制两条线的图形。

In this case, let’s make a table of values for each line.
::在此情况下,让我们为每行绘制一个数值表。

Line 1: $\begin{align*}y = 3x - 5\end{align*}$
::第1行:y=3x-5

$\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & \text{-}2 \\\hline 2 & 1 \\\hline \end{array}\end{align*}$
::xy1-221xy1-221

Line 2: $\begin{align*}y = -2x + 5\end{align*}$
::第2行 : y2x+5

$\begin{align*}\begin{array}{|c|c|} \hline x & y \\\hline 1 & 3 \\\hline 2 & 1 \\\hline \end{array}\end{align*}$
::xy1321 xy1321

The solution to the system is given by the intersection point of the two lines. The graph shows that the lines intersect at point (2, 1). So the solution is $\begin{align*}\mathbf{x = 2, \ y = 1}\end{align*}$ or (2, 1).
::系统解决方案由两行的交叉点给出。图表显示, 线在点(2, 1) 交叉, 所以答案是x=2, y=1or (2, 1) 。

Solving a System of Equations Using a Graphing Calculator
::使用图形计算计算器解决公式系统

As an alternative to graphing by hand, you can use a graphing calculator to find or check solutions to a system of equations.
::作为用手绘制图的替代方法,您可以使用图形计算器来寻找或检查方程式系统的解决方案。

Solve the system of equations using a graphing calculator.
::使用图形计算计算器解决方程式系统。

$\begin{align*}\begin{cases} x - 3y = 4\\ 2x + 5y = 8 \end{cases}\end{align*}$
::{x-3y=42x+5y=8

To input the equations into the calculator, you need to rewrite them in slope-intercept form (that is, $\begin{align*}y = mx + b\end{align*}$ form).
::要将方程式输入计算器, 您需要以斜度界面( 即 y=mx+b 窗体) 重写这些方程式。

$\begin{align*}x - 3y = 4 \quad &\Rightarrow \quad y = \frac{1}{3}x - \frac{4}{3}\!\\ \\ 2x + 5y = 8 \quad &\Rightarrow \quad y = - \frac{2}{5}x + \frac{8}{5}\end{align*}$
::x-3y=4y=13x-432x+5y=8y25x+85

Press the [y=] button on the graphing calculator and enter the two functions as:
::按下图形计算计算器上的 [y=] 按钮,并输入两个函数如下:

$\begin{align*}Y_1 & = \frac{x}{3} - \frac{4}{3}\\ Y_2 & = \frac{-2x}{5} + \frac{8}{5}\end{align*}$
::Y1=x3- 43Y2=2x5+85

Now press [GRAPH] . Here’s what the graph should look like on a TI-83 family graphing calculator with the window set to $\begin{align*}-5 \le x \le 10\end{align*}$ and $\begin{align*}-5 \le y \le 5\end{align*}$ .
::现在按下 [GRAPH] 键。图表在 TI- 83 家庭图形计算器上应该是什么样子, 窗口设置为 5 x 10 和 5 y 5 5 。

There are a few different ways to find the intersection point.
::找到交叉点有几种不同的方法。

Option 1 : Use [TRACE] and move the cursor with the arrows until it is on top of the intersection point. The values of the coordinate point will be shown on the bottom of the screen. The second screen above shows the values to be $\begin{align*}X = 4.0957447\end{align*}$ and $\begin{align*}Y = 0.03191489\end{align*}$ .
::选项1:使用 [TRACE] 并用箭头移动光标, 直至它位于交叉点的顶端。坐标点的值将在屏幕底部显示。以上第二个屏幕显示的值为 X= 4. 0957447 和 Y= 0.03191489。

Use the [ZOOM] function to zoom into the intersection point and find a more accurate result. The third screen above shows the system of equations after zooming in several times. A more accurate solution appears to be $\begin{align*}X = 4\end{align*}$ and $\begin{align*}Y = 0\end{align*}$ .
::使用 [ZOOM] 函数缩放到交叉点并找到更准确的结果。上面的第三个屏幕显示多次缩放后的方程式系统。更准确的解决方案似乎是 X=4 和 Y=0 。

Option 2 : Look at the table of values by pressing [2nd] [GRAPH] . The first screen below shows a table of values for this system of equations. Scroll down until the $\begin{align*}Y-\end{align*}$ values for the two functions are the same. In this case this occurs at $\begin{align*}X = 4\end{align*}$ and $\begin{align*}Y = 0\end{align*}$ .
::选项2:按下 [第二 [GRAPH] 查看数值表。以下第一个屏幕显示此等式系统的数值表。向下滚动,直到两个函数的Y-值相同。在这种情况下,在 X=4 和 Y=0 时发生。

(Use the [TBLSET] function to change the starting value for your table of values so that it is close to the intersection point and you don’t have to scroll too long. You can also improve the accuracy of the solution by setting the value of the $\begin{align*}\Delta\end{align*}$ Table smaller.)
:使用 [TBLSET] 函数来改变数值表的起始值, 以便它接近交叉点, 您不必滚动太长。您也可以通过设定表的较小值来提高解决方案的准确性。 )

Option 3 : Using the [2nd] [TRACE] function gives the second screen shown above.
::备选案文3:使用[第2 [TRACE]函数使上文显示的第二个屏幕出现。

Scroll down and select “ intersect .”
::向下滚动并选择“交叉 ” 。

The calculator will display the graph with the question [FIRSTCURVE] ? Move the cursor along the first curve until it is close to the intersection and press [ENTER] .
::计算器将显示图中的问题 [FIRSTCURVE] 。光标将沿着第一个曲线移动, 直到它接近交叉点并按 [ENTER] 。

The calculator now shows [SECONDCURVE] ?
::计算器现在显示 [SECONDCURVE]?

Move the cursor to the second line (if necessary) and press [ENTER] .
::将光标移动到第二行(如有必要)并按 [ENTER] 键。

The calculator displays [GUESS] ?
::计算器显示?

Press [ENTER] and the calculator displays the solution at the bottom of the screen (see the third screen above).
::按 [ENTER] 键和计算器在屏幕底部显示溶液(见上文第三个屏幕)。

The point of intersection is $\begin{align*}X = 4\end{align*}$ and $\begin{align*}Y = 0\end{align*}$ . Note that with this method, the calculator works out the intersection point for you, which is generally more accurate than your own visual estimate.
::交叉点是 X=4 和 Y=0。请注意, 使用此方法, 计算器为您绘制交叉点, 通常比您自己的视觉估计更准确。

Solve Real-World Problems Using Graphs of Linear Systems
::使用线性系统图解解决现实世界问题

Consider the following problem:
::考虑以下问题:

Peter and Nadia like to race each other. Peter can run at a speed of 5 feet per second and Nadia can run at a speed of 6 feet per second. To be a good sport, Nadia likes to give Peter a head start of 20 feet. How long does Nadia take to catch up with Peter? At what distance from the start does Nadia catch up with Peter?
::彼得和纳迪亚喜欢相互竞争。彼得每秒可以跑5英尺的速度, 纳迪亚每秒可以跑6英尺的速度。作为一项好运动, Nadia喜欢让Peter先跑20英尺。 Nadia要多久才能赶上Peter? Nadia从什么时候开始会赶上Peter?

S tart by drawing a sketch. Here’s what the race looks like when Nadia starts running . C all this time $\begin{align*}t = 0\end{align*}$ .
::首先绘制一张草图。当Nadia开始运行时, 比赛的景象是这样的。拨打这个时间 t=0 。

Now, define two variables in this problem:
::现在,在此问题上定义两个变量 :

$\begin{align*}t =\end{align*}$ the time from when Nadia starts running
::t= Nadia 开始运行的时间

$\begin{align*}d =\end{align*}$ the distance of the runners from the starting point .
::d= 赛跑者与起点的距离。

Since there are two runners, there need to be equations for each of them. That will be the system of equations for this problem.
::由于有两名选手,他们每人需要有方程式。这将是这一问题的方程式系统。

For each equation, use the formula : $\begin{align*}\text{distance} = \text{speed}\times \text{time}\end{align*}$
::对于每个方程,使用公式: 距离= 速度x 时间

Nadia’s equation: $\begin{align*}d = 6t\end{align*}$
::Nadia的方程: d=6t

Peter’s equation: $\begin{align*}d = 5t + 20\end{align*}$
::彼得的方程: d=5t+20

(Remember that Peter was already 20 feet from the starting point when Nadia started running.)
:记得当Nadia开始奔跑时 Peter已经离起点20英尺了。 )

Let’s graph these two equations on the same coordinate axes.
::让我们用相同的坐标轴来图解这两个方程式。

Time should be on the horizontal axis since it is the independent variable . Distance should be on the vertical axis since it is the dependent variable.
::时间应该放在水平轴上, 因为它是独立的变量。距离应该放在垂直轴上, 因为它是独立的变量。

We can use any method for graphing the lines, but in this case we’ll use the slope– intercept method since it makes more sense physically.
::我们可以使用任何方法来绘制线条图, 但在此情况下我们会使用斜坡—— 拦截法,

To graph the line that describes Nadia’s run, start by graphing her equation's $\begin{align*}y-\end{align*}$ intercept: (0, 0). (If you don’t understand why this is the $\begin{align*}y-\end{align*}$ intercept, try plugging in the test-value of $\begin{align*}x = 0\end{align*}$ .)
::绘制描述 Nadia 运行的线条, 首先绘制她的公式 y - interfict: (0, 0) (如果你不明白为什么这是 y - interfict, 请尝试插入 x=0 的测试值。)

The tells us that Nadia runs 6 feet every one second, so another point on the line is (1, 6). Connecting these points gives us Nadia’s line:
::告诉我们Nadia每秒跑6英尺, 线上的另一点是(1,6),

To graph the line that describes Peter’s run, start with his $\begin{align*}y-\end{align*}$ intercept. In this case, this is the point (0, 20).
::要绘制描述 Peter 运行的线条, 请从 Y - intercut 开始。在这种情况下, 这就是点 ( 0, 20 ) 。

The slope tells us that Peter runs 5 feet every one second, so another point on the line is (1, 25). Connecting these points gives Peter’s line:
::斜坡告诉我们,彼得每秒跑5英尺,所以线上的另一个点是(1,25)。

In order to find when and where Nadia and Peter meet, graph both lines on the same graph and extend the lines until they cross. The crossing point is the solution to this problem.
::为了找到Nadia和Peter相遇的时间和地点,请用同一图表绘制两条线的图表,并将两条线延伸至经过时。过境点是解决这个问题的办法。

The graph shows that Nadia and Peter meet 20 seconds after Nadia starts running, and 120 feet from the starting point.
::该图显示,Nadia和Peter在Nadia开始运行20秒后相遇,距离起点120英尺。

These examples are great at demonstrating that the solution to a system of linear equations is the point at which the lines intersect. This is, in fact, the greatest strength of the graphing method - it offers a very visual representation of a system of equations and its solution. You can also see that finding the solution from a graph requires very careful graphing of the lines, and is really only practical when you’re sure that the solution gives integer values for $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ . Often , this method can only offer approximate solutions to systems of equations, so you need to use other methods to get an exact solution.
::这些例子可以很好地证明线性方程式系统的解决方案是线性方程式相交点。事实上,这是图形化方法的最大强点 — — 它非常直观地展示了方程式及其解决方案的系统。您还可以看到,从图形中找到解决方案需要非常仔细地绘制线性方程式的图形,而且只有在您确信该解决方案给出 x 和 y 的整数值时,该方法才真正实用。通常,这种方法只能为方程式系统提供大致的解决方案,所以您需要使用其他方法来找到准确的解决方案。

Example
::示例示例示例示例

Example 1
::例1

Solve the following system of equations by graphing:
::以图示方式解决下列方程式系统:

$\begin{align*}\begin{cases}2x + 3y = 6\\ \ \ 4x - y = \text{-}2 \end{cases}\end{align*}$
::{2x+3y=6 4x-y=-2

Since the equations are in standard form , graph them by finding the $\begin{align*}x-\end{align*}$ and $\begin{align*}y-\end{align*}$ intercepts of each of the lines.
::由于方程式是标准格式,所以通过查找每行的 x 和 y- 截面来绘制图形。

Line 1: $\begin{align*}2x + 3y = 6\end{align*}$
::第1:2x+3y=6行

$\begin{align*}x-\end{align*}$ intercept: set $\begin{align*}y = 0 \text{, giving } 2x = 6 \Rightarrow x = 3\end{align*}$ so the intercept is (3, 0)
::x- 截取: 设置 Y=0, 给 2x= 6 x=3 所以拦截是 (3, 0)

$\begin{align*}y-\end{align*}$ intercept: set $\begin{align*}x = 0 \text{, giving } 3y = 6 \Rightarrow y = 2\end{align*}$ so the intercept is (0, 2)
::y- 截取: 设置 x=0, 给 3y=6\\\\ y=2 所以拦截是 (0, 2)

Line 2: $\begin{align*}\text{-}4x + y = 2\end{align*}$
::第2行:4x+y=2

$\begin{align*}x-\end{align*}$ intercept: set $\begin{align*}y = 0 \text{, giving } \text{-}4x = 2 \Rightarrow x = \text{-} \frac{1}{2}\end{align*}$ so the intercept is $\begin{align*}\left ( \text{-} \frac{1}{2}, 0 \right )\end{align*}$
::x- 截取 : 设置 Y=0 , 给 - 4x=2\\\\\ x= 12, 所以拦截是 (12,0)

$\begin{align*}y-\end{align*}$ intercept: set $\begin{align*}x = 0 \text{, giving } y = 2\end{align*}$ so the intercept is (0, 2)
::y- 拦截: set x=0, give y=2 所以拦截是 (0, 2)

The graph shows that the lines intersect at (0, 2). Therefore , the solution to the system of equations is $\begin{align*}\mathbf{x = 0, \ y = 2.}\end{align*}$
::该图显示,线条在 0, 2 点交叉,因此,对等方程式系统的解决方案是x=0, y =2。

Review
::回顾

Determine which ordered pair satisfies each system of linear equations.
::确定哪对定购单符合每一线性方程系统。

$\begin{align*}\begin{cases}y = 3x - 2\!\\ y = \text{-}x\end{cases}\end{align*}$

(1, 4)

(2, 9)

$\begin{align*}\left (\frac{1}{2}, \ \text{-}\frac{1}{2} \right )\\ \ \\\end{align*}$

::{y=3x-2y=-x(1,4,2,9)(12,12)

$\begin{align*}\begin{cases}y = 2x - 3\\ y = x + 5 \end{cases}\end{align*}$

(8, 13)

(-7, 6)

(0, 4)

::{y=2x-3y=x+5(8,13)(7,6)(0,4)

$\begin{align*}\\ \begin{cases}\ \ 2x + y = 8\\ 5x + 2y = 10\end{cases}\end{align*}$

(-9, 1)

(-6, 20)

(14, 2)

::{2x+y=85x+2y=10(9、9、1)(6、20)(14、2)

$\begin{align*}\\ \begin{cases}3x + 2y = 6\\ y = \frac{1}{2}x - 3 \end{cases}\end{align*}$

$\begin{align*}\left (3, \frac{-3}{2} \right )\end{align*}$

(-4, 3)

$\begin{align*}\left (\frac{1}{2}, \ 4 \right )\\ \ \\\end{align*}$

::{3x+2y=6y=12x-3(3,-32(4,4,3))(12,4)

$\begin{align*}\begin{cases}2x - y = 10\\ 3x + y = \text{-}5 \end{cases}\end{align*}$

(4, -2)

(1, -8)

(-2, 5)

::{2x-y=103x+y=5 (4,-2) (4,-2) (1,8) (2,5)

Solve the following systems using the graphing method.
::使用图形绘制方法解决下列系统。

$\begin{align*}\begin{cases}y = x + 3\\ y = \text{-}x + 3 \end{cases}\\ \ \\\end{align*}$
::{y=x+3y=-x+3=-x+3

$\begin{align*}\begin{cases}y = 3x - 6\\ y = \text{-}x + 6 \end{cases}\\ \ \\\end{align*}$
::{y=3x-6y=-x+6 {y=3x-6y=-x+6

$\begin{align*}\begin{cases}2x = 4\\ \ \ y = \text{-}3 \end{cases}\\ \ \\\end{align*}$
::{2x=4y=3} {2x=4y=3

$\begin{align*}\begin{cases}y = \text{-}x + 5\\ \text{-}x + y = 1 \end{cases}\\ \ \\\end{align*}$
::{y=- x+5- x+y=1

$\begin{align*}\begin{cases}\ \ x + 2y = 8\\ 5x + 2y = 0 \end{cases}\\ \ \\\end{align*}$
::{ x+2y=85x+2y=0

$\begin{align*}\begin{cases} 3x + 2y = 12\\ \ \ 4x - y = 5 \end{cases}\\ \ \\\end{align*}$
::{3x+2y=12 4x-y=5

$\begin{align*}\begin{cases}5x + 2y = \text{-}4\\ x - y = 2 \end{cases}\\ \ \\\end{align*}$
::{5x+2y=4x-y=2

$\begin{align*}\begin{cases} 2x + 4 = 3y\\ x - 2y + 4 = 0 \end{cases}\\ \ \\\end{align*}$
::{2x+4=3yx-2y+4=0

$\begin{align*}\begin{cases}y = \frac{1}{2}x - 3\\ 2x - 5y = 5 \end{cases}\\ \ \\\end{align*}$
::{y=12x-32x-5y=5

$\begin{align*}\begin{cases}y = 4\\ x = 8 - 3y \end{cases}\\ \ \\\end{align*}$
::{y=4x=8-3y

Try to solve the following system using the graphing method:

$\begin{align*}\begin{cases}y = \frac{3}{5}x + 5\\ y = \text{-}2x - \frac{1}{2} \end{cases}\end{align*}$

What does it look like the $\begin{align*}x-\end{align*}$ coordinate of the solution should be?
::解决办法的X协调应该是什么样子?

Does that coordinate really give the same $\begin{align*}y-\end{align*}$ value when you plug it into both equations?
::当您将其插入两个方程式时, 此坐标是否真的给与相同的 Y - 值 ?

Why is it difficult to find the real solution to this system?
::为什么很难找到这个系统的真正解决办法?

::尝试用图形化方法解析以下系统 : {y= 35x+5y=-2x-12 : 解决方案的 X - 坐标应该是什么? 该坐标是否在将它插入两个方程式时给与相同的 Y - 值 ? 为什么很难找到这个系统的真正解决方案 ?

Try to solve the following system using the graphing method. Use a grid with $\begin{align*}x-\end{align*}$ values and $\begin{align*}y-\end{align*}$ values ranging from -10 to 10 .

$\begin{align*}\begin{cases}y = 4x + 8\\ y = 5x + 1 \end{cases}\end{align*}$

Do these lines appear to intersect?
::这些线条似乎交叉吗?

Based on their equations, are they parallel?
::根据他们的方程式,它们平行吗?

What needs to be done to find their intersection point?
::需要做些什么才能找到他们的交叉点?

::尝试使用图形化方法解析以下系统。使用从 - 10 到 10. {y= 4x+8y=5x+1 的有x- 值和y- 值的网格。这些线线似乎交叉吗 ? 基于它们的方程式, 它们是否平行 ? 要找到它们的交叉点需要做什么 ?

Try to solve the following system using the graphing method. Use a grid with $\begin{align*}x-\end{align*}$ values and $\begin{align*}y-\end{align*}$ values ranging from -10 to 10 .

$\begin{align*}\begin{cases}y = \frac{1}{2}x + 4\\ y = \frac{4}{9}x + \frac{9}{2}\end{cases}\end{align*}$

Can you tell exactly where the lines cross?
::你能分辨出线条到底交叉到哪里去吗?

What would we have to do to make it clearer?
::我们要做些什么才能使它更清楚?

::尝试使用图形化方法解析以下系统。使用从 - 10 到 10. {y= 12x+4y= 49x+92 的有x- values andy- y- value 的网格。您能否辨别行的交叉位置 ? 需要做些什么才能弄清楚 ?

Solve the following problems by using the graphing method.
::使用图形绘制方法解决下列问题。

Mary’s car has broken down and it will cost her $1200 to get it fixed, or, for $4500, she can buy a new and more efficient car instead. Her present car uses about $2000 worth of gas per year, while gas for the new car would cost about $1500 per year. After how many years would the total cost of fixing the car equal the total cost of replacing it?
::玛丽的汽车已经坏了,修车成本为1200美元,或者4500美元,她可以买一辆新的、更有效率的汽车。她现在的汽车每年消耗大约2000美元的汽油,而新车的汽油每年花费大约1500美元。修车的总成本要花多少年才能与更换汽车的总成本相等?

A tortoise and hare decide to race 30 feet. The hare, being much faster, decides to give the tortoise a 20 foot head start. The tortoise runs at 0.5 feet/sec and the hare runs at 5.5 feet per second. How long until the hare catches the tortoise?
::一只乌龟和野兔决定赛跑30英尺。兔子跑得更快,决定让乌龟领先20英尺。乌龟以0.5英尺/秒的速度运行,野兔以每秒5.5英尺的速度运行。兔子要多久才能抓住乌龟?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

线性系统图图

章节大纲

Graphs of Linear Systems
::线性系统图图

Determine Whether an Ordered Pair is a Solution to a System of Equations
::确定有秩序对等是否是公式体系的解决方案

Finding the Solution to a System of Equations
::寻找公式体系的解决方案

Determine the Solution to a Linear System by Graphing
::通过图形绘制确定线性系统的解决方案

Solving a System of Equations by Graphing
::通过图形绘制解决方平系统

Solving a System of Equations Using a Graphing Calculator
::使用图形计算计算器解决公式系统

Solve Real-World Problems Using Graphs of Linear Systems
::使用线性系统图解解决现实世界问题

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)

章节大纲

Graphs of Linear Systems ::线性系统图图

Determine Whether an Ordered Pair is a Solution to a System of Equations ::确定有秩序对等是否是公式体系的解决方案

Finding the Solution to a System of Equations ::寻找公式体系的解决方案

Determine the Solution to a Linear System by Graphing ::通过图形绘制确定线性系统的解决方案

Solving a System of Equations by Graphing ::通过图形绘制解决方平系统

Solving a System of Equations Using a Graphing Calculator ::使用图形计算计算器解决公式系统

Solve Real-World Problems Using Graphs of Linear Systems ::使用线性系统图解解决现实世界问题

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Graphs of Linear Systems
::线性系统图图

Determine Whether an Ordered Pair is a Solution to a System of Equations
::确定有秩序对等是否是公式体系的解决方案

Finding the Solution to a System of Equations
::寻找公式体系的解决方案

Determine the Solution to a Linear System by Graphing
::通过图形绘制确定线性系统的解决方案

Solving a System of Equations by Graphing
::通过图形绘制解决方平系统

Solving a System of Equations Using a Graphing Calculator
::使用图形计算计算器解决公式系统

Solve Real-World Problems Using Graphs of Linear Systems
::使用线性系统图解解决现实世界问题

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)