Course: 9.4 找到两种变量中等式的解决办法

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找到两个变量中等式的解决方案

Mr. Brown, a seventh grade teacher, calls the Science Museum to find out the cost of taking his class to the Omni Theater to see a film on the rainforest. He receives the following information.
::Brown先生是一名七年级教师,他给科学博物馆打电话,要求他了解去奥姆尼剧院看雨林电影的费用。他收到以下信息。

The cost of a ticket is $5.00 plus a $2.00 service fee per ticket. There are 22 students in his class and he could have anywhere from 22 to 18 students go based on the number of students absent. Now Mr. Brown has to do the math to figure out the cost of taking the students to see the film.
::票价为5.00美元加每张票的2美元服务费,班上有22名学生,根据缺席学生人数,他可以让22至18名学生到任何地方。现在布朗先生必须计算一下带学生去看电影的费用。

He knows that he will have to use some type of an equation and a table to record the cost based on the different number of students that may attend.
::他知道,他将不得不使用某种方程和一张表格,根据可能参加的学生人数的不同记录费用。

In this concept, you will learn to find solutions for equations in two variables.
::在此概念中,您将学会找到两个变量方程式的解决方案。

Equations
::平数

Function rules can be written using words or symbols. When symbols are used the function rule takes on the form of an equation. Remember, an equation is a statement of equality between two or more quantities. The equation $\begin{align*}x + y = 5\end{align*}$ means there are values for ‘ $\begin{align*}x\end{align*}$ ’ and ‘ $\begin{align*}y\end{align*}$ ’ that would make the statement true. This equation contains two variables, ‘ $\begin{align*}x\end{align*}$ ’ and ‘ $\begin{align*}y\end{align*}$ ’, and any solution can be written as an ordered pair , $\begin{align*}(x, y)\end{align*}$ .
::函数规则可以用文字或符号来写。当使用符号时, 函数规则以等式的形式出现。记住, 等式是两个或两个以上数量之间的平等声明。等式 x+y=5 表示有 'x ' 和`y ' 的值, 使语句成为事实。这个等式包含两个变量, `x ' 和`y ' , 任何解决方案都可以以定购配对的形式写入 (x,y) 。

Let’s look at some solutions for the equation.
::让我们来看看这个等式的解决方案。

\begin{aligned} x + y = 5 \end{aligned}

$\begin{align*}x + y = 5\end{align*}$
::x+y=5 x+y=5

$\begin{align*}3+2=5\end{align*}$ which means $\begin{align*}x=3\end{align*}$ and $\begin{align*}y=2\end{align*}$ . The solution can be written as $\begin{align*}(3,2)\end{align*}$ .
::3+2=5 表示 x=3 和 y=2 。解决方案可以写为 (3, 2) 。

$\begin{align*}7+(-2)=5\end{align*}$ which means $\begin{align*}x=7\end{align*}$ and $\begin{align*}y=-2\end{align*}$ . The solution can be written as $\begin{align*}(7,-2)\end{align*}$ .
::7+(-2)=5 表示 x=7 和 y2. 解决方法可以写为 (7,-2) 。

$\begin{align*}0+5=5\end{align*}$ which means $\begin{align*}x=0\end{align*}$ and $\begin{align*}y=5\end{align*}$ . The solution can be written as $\begin{align*}(0,5)\end{align*}$ .
::0+5=5 表示 x=0 和 y=5 。解决方案可以写为 (0, 5) 。

The number of ordered pairs that will make the given equation a true statement of equality is endless.
::使特定方程式成为真正的平等声明的定购配对数量是无穷无尽的。

Look at the following equation with two variables ‘ $\begin{align*}x\end{align*}$ ’ and ‘ $\begin{align*}y\end{align*}$ .’
::以两个变量 'x ' 和`y' 来查看下面的方程式。

\begin{aligned} 2 x + y = 12 \end{aligned}

$\begin{align*}2x+y=12\end{align*}$
::2x+y=12

This is another equation that has many ordered pairs that make it a true statement. Write four ordered pairs for the given equation.
::这是另一个配方, 有很多定购的对, 使它成为真实声明。为给定方程写四对定购的对。

\begin{aligned} \begin{array}{rcl} 2 x + y & = & 12 \\ 2 (1) + (10) & = & 12 The ordered pair is (1, 10) . \\ 2 + 10 & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 2x+y&=&12\\ 2(1)+(10)&=&12 \qquad \text{The ordered pair is } (1,10). \\ 2+10&=&12\\ 12&=&12 \end{array}\end{align*}$
::2x+y=122(1)+(10)=12 订购的一对(1,10) 2+10=1212=12

\begin{aligned} \begin{array}{rcl} 2 x + y & = & 12 \\ 2 (2) + (8) & = & 12 The ordered pair is (2, 8) . \\ 4 + 8 & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 2x+y&=&12\\ 2(2)+(8)&=&12 \qquad \text{The ordered pair is } (2,8). \\ 4+8&=&12\\ 12&=&12 \end{array}\end{align*}$
::2x+y=122(2)+(8)=12 订购的一对是(2,8),4+8=1212=12。

\begin{aligned} \begin{array}{rcl} 2 x + y & = & 12 \\ 2 (3) + (6) & = & 12 The ordered pair is (3, 6) . \\ 6 + 6 & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 2x+y&=&12\\ 2(3)+(6)&=&12 \qquad \text{The ordered pair is } (3,6). \\ 6+6&=&12\\ 12&=&12 \end{array}\end{align*}$
::2x+y=122(3)+(6)=12 订购的对(3,6)6+6=1212=12)

\begin{aligned} \begin{array}{rcl} 2 x + y & = & 12 \\ 2 (5) + (2) & = & 12 The ordered pair is (5, 2) . \\ 10 + 2 & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 2x+y&=&12\\ 2(5)+(2)&=&12 \qquad \text{The ordered pair is } (5,2). \\ 10+2&=&12\\ 12&=&12 \end{array}\end{align*}$
::2x+y=122(5)+(2)=12 订购的一对是 (5,2,10+2=12=12)

The given equation is written in the form $\begin{align*}Ax+By=C\end{align*}$ where $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ are coefficients of the variables $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ respectively and $\begin{align*}C\end{align*}$ is a constant . An equation written in the form $\begin{align*}Ax+By=C\end{align*}$ is said to be written in standard form . When an equation is written in standard form, the coefficient ‘ $\begin{align*}A\end{align*}$ ’ of the variable ‘ $\begin{align*}x\end{align*}$ ’ must be a positive value.
::给定方程式以 Ax+By=C 格式写成, 其中 A 和 B 是变量 x 和 y 的系数, C 是一个常数。以 Ax+By=C 格式写成的方程式, 据说以标准格式写成。当方程式以标准格式写成时, 变量 `x ' 的系数 `A ' 必须是正值。

Another way to write the given equation with two variables ‘ $\begin{align*}x\end{align*}$ ’ and ‘ $\begin{align*}y\end{align*}$ ’ is to write it in function form as $\begin{align*}y=-2x+12\end{align*}$ . The variable ‘ $\begin{align*}y\end{align*}$ ’ has been written as a function of $\begin{align*}-2x+12\end{align*}$ which means that the value of the output number ‘ $\begin{align*}y\end{align*}$ ’ will depend upon the value substituted into the function for ‘ $\begin{align*}x\end{align*}$ ’ variable. If is used such that $\begin{align*}y=f(x)\end{align*}$ then the function form of the equation will be $\begin{align*}f(x)=-2x+12\end{align*}$ . Whatever form is used, the value of the output number will change as the value of the input number changes.
::以两个变量 ' x ' 和 'y ' 写入给定方程式的另一种方式是以 y\\\ 2x+12 的形式以函数形式写入它。变量`y ' 被写成为 ~2x+12 的函数,这意味着输出号`y ' 的值将取决于 y 的值。如果使用 y=f(x) 的值,则方程式的函数形式将是 f(x)\\\\ 2x+12。无论使用哪种形式,输出号的值将随着输入号的值变化而变化。

Use the given equation, written in function form, to complete the following table.
::使用以函数格式写成的给定方程式来填写下表。

\begin{aligned} y = 3 x - 5 \end{aligned}

$\begin{align*}y=3x-5\end{align*}$
::y= 3x- 5

$\begin{align}x\end{align}$	$\begin{align}y\end{align}$
1	-2
2	1
3	4
4	7
5	10

First, substitute ‘1’ into the function for the variable ‘ $\begin{align*}x\end{align*}$ .’
::首先,将变量 'x. ' 的函数替换为 '1 ' 。

\begin{aligned} \begin{array}{rcl} y & = & 3 x - 5 \\ y & = & 3 (1) - 5 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} y&=&3x-5\\ y&=&3(1)-5 \end{array}\end{align*}$
::y=3x-5y=3(1)-5

Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。

\begin{aligned} \begin{array}{rcl} y & = & 3 (1) - 5 \\ y & = & 3 - 5 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} y&=&3(1)-5\\ y&=&3-5 \end{array}\end{align*}$
::y=3(1)-5y=3-5

Next, perform the subtraction on the right side of the equation.
::下一步,在方程右侧执行减法。

\begin{aligned} \begin{array}{rcl} y & = & 3 - 5 \\ y & = & - 2 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} y&=&3-5\\ y&=&-2 \end{array}\end{align*}$
::y=3 - 5y2

Then, write the answer in the table.
::那就把答案写在桌子上

The answer is -2.
::答案是 -2

Repeat the above process for the remaining values of the variable ‘ $\begin{align*}x\end{align*}$ .’
::对于变量 'x.' 的剩余值,重复上述进程。

\begin{aligned} \begin{array}{rcl} x & = & 2 \\ y & = & 3 x - 5 \\ y & = & 3 (2) - 5 \\ y & = & 6 - 5 \\ y & = & 1 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&2\\ y&=&3x-5\\ y&=&3(2)-5\\ y&=&6-5\\ y&=&1\\ \end{array}\end{align*}$
::x=2y=3x-5y=3(2)-5y=6-5y=1

The answer is 1.
::答案是 1 。

\begin{aligned} \begin{array}{rcl} x & = & 3 \\ y & = & 3 x - 5 \\ y & = & 3 (3) - 5 \\ y & = & 9 - 5 \\ y & = & 4 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&3\\ y&=&3x-5\\ y&=&3(3)-5\\ y&=&9-5\\ y&=&4\\ \end{array}\end{align*}$
::x=3y=3x-5y=3(3)-5y=9-5y=4

The answer is 4.
::答案是4。

\begin{aligned} \begin{array}{rcl} x & = & 4 \\ y & = & 3 x - 5 \\ y & = & 3 (4) - 5 \\ y & = & 12 - 5 \\ y & = & 7 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&4\\ y&=&3x-5\\ y&=&3(4)-5\\ y&=&12-5\\ y&=&7\\ \end{array}\end{align*}$
::x=4y=3x-5y=3(4)-5y=12-5y=7

The answer is 7.
::答案是7个

\begin{aligned} \begin{array}{rcl} x & = & 5 \\ y & = & 3 x - 5 \\ y & = & 3 (5) - 5 \\ y & = & 15 - 5 \\ y & = & 10 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&5\\ y&=&3x-5\\ y&=&3(5)-5\\ y&=&15-5\\ y&=&10\\ \end{array}\end{align*}$
::x=5y=3x-5y=3(5)-5y=15-5y=10

The answer is 10.
::答案是10个

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mr. Brown and the rainforest movie. He needs to figure out the cost of taking the students to see the rainforest movie. He can write an equation in function form to represent the information he received from the Science Museum.
::早些时候,有人给了你一个关于布朗先生和雨林电影的问题。他需要弄清楚带学生去看雨林电影的成本。他可以以功能形式写一个方程式来代表他从科学博物馆得到的信息。

First, write the information from the Science Museum.
::首先,写科学博物馆的资料。

Each ticket costs $5.00 plus a $2.00 fee per ticket.
::每张票价为5.00美元,加上每张票价为2.00美元。

Next, write an equation in function form to represent the information.
::下一步,在函数格式中写一个方程式以表示信息。

Let ‘ $\begin{align*}x\end{align*}$ ’ represent the number of tickets being purchased and let ‘ $\begin{align*}y\end{align*}$ ’ represent the total cost for the number of tickets purchased.
::让`x ' 代表购买的机票数量, 让`y ' 代表购买的机票总成本。

\begin{aligned} y = 5 x + 2 \end{aligned}

$\begin{align*}y=5x+2\end{align*}$
::y=5x+2 y=5x+2

Next, draw a table to record the cost for taking 18, 19, 20, 21 or 22 students to the movie. Each student must purchase a ticket.
::接下来,绘制一张表格,记录18、19、20、21或22名学生观看电影的费用。每个学生必须购买一张票。

Number of Tickets ( $\begin{align}x\end{align}$ )	Total Cost ( $\begin{align}y\end{align}$ )
18	$92.00
19	$97.00
20	$102.00
21	$107.00
22	$112.00

Use the equation to calculate the cost for each group of students to see the movie.
::使用方程式计算每组学生看电影的费用。

\begin{aligned} \begin{array}{rcl} x & = & 18 \\ y & = & 5 x + 2 \\ y & = & 5 (18) + 2 Substitute x = 18 into the equation . \\ y & = & 90 + 2 Perform the multiplication to clear the parenthesis . \\ y & = & $ 92.00 Simplify the right side of the equation . \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&18\\ y&=&5x+2\\ y&=&5(18)+2 \qquad \text{Substitute } x=18 \text{ into the equation}.\\ y&=&90+2 \qquad \quad \ \text{Perform the multiplication to clear the parenthesis}.\\ y&=&\$92.00 \qquad \quad \ \text{Simplify the right side of the equation}. \end{array}\end{align*}$
::x=18y=5x+2y=5(18)+2替代 x=18进入方程式。y=90+2 执行乘法清除括号。y=92.00 简化方程式右侧。

Repeat the process above to calculate the cost for the remaining ‘ $\begin{align*}x\end{align*}$ ’ values.
::重复以上计算剩余 " x " 值的成本的过程。

\begin{aligned} \begin{array}{rcl} x & = & 19 \\ y & = & 5 x + 2 \\ y & = & 5 (19) + 2 \\ y & = & 95 + 2 \\ y & = & $ 97.00 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&19\\ y&=&5x+2\\ y&=&5(19)+2\\ y&=&95+2\\ y&=&\$97.00 \end{array}\end{align*}$
::x=19y=5x+2y=5(19)+2y=95+2y=97.00美元

\begin{aligned} \begin{array}{rcl} x & = & 20 \\ y & = & 5 x + 2 \\ y & = & 5 (20) + 2 \\ y & = & 100 + 2 \\ y & = & $ 102.00 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&20\\ y&=&5x+2\\ y&=&5(20)+2\\ y&=&100+2\\ y&=&\$102.00 \end{array}\end{align*}$
::x=20y=5x+2y=5(20)+2y=100+2y=102.00美元

\begin{aligned} \begin{array}{rcl} x & = & 21 \\ y & = & 5 x + 2 \\ y & = & 5 (21) + 2 \\ y & = & 105 + 2 \\ y & = & $ 107.00 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&21\\ y&=&5x+2\\ y&=&5(21)+2\\ y&=&105+2\\ y&=&\$107.00 \end{array}\end{align*}$
::x=21y=5x+2y=5(21)+2y=105+2y=107.00美元

\begin{aligned} \begin{array}{rcl} x & = & 22 \\ y & = & 5 x + 2 \\ y & = & 5 (22) + 2 \\ y & = & 110 + 2 \\ y & = & $ 112.00 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&22\\ y&=&5x+2\\ y&=&5(22)+2\\ y&=&110+2\\ y&=&\$112.00 \end{array}\end{align*}$
::x=22y=5x+2y=5(22)+2y=110+2y=112.00美元

Example 2
::例2

Write the following equation in standard form.
::以标准形式写下以下方程。

\begin{aligned} 7 y - 9 = 4 x \end{aligned}

$\begin{align*}7y-9=4x\end{align*}$
::7y-9=4x

First, write the standard form of an equation.
::首先,写一个方程的标准格式。

\begin{aligned} A x + B y = C \end{aligned}

$\begin{align*}Ax+By=C\end{align*}$
::Ax+By=C 轴+By=C

Next, add 9 to both sides of the equation and simplify the equation.
::接下来,在方程的两侧加上9,简化方程。

\begin{aligned} \begin{array}{rcl} 7 y - 9 & = & 4 x \\ 7 y - 9 + 9 & = & 4 x + 9 \\ 7 y & = & 4 x + 9 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 7y-9&=&4x\\ 7y-9+9&=&4x+9\\ 7y&=&4x+9 \end{array}\end{align*}$
::7-9=4x7y-9+9=4x-97y=4x+9

Next, subtract $\begin{align*}4x\end{align*}$ from both sides of the equation and simplify the equation.
::接下来,从方程两侧减去4x,简化方程。

\begin{aligned} \begin{array}{rcl} 7 y & = & 4 x + 9 \\ 7 y - 4 x & = & 4 x - 4 x + 9 \\ 7 y - 4 x & = & 9 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 7y&=&4x+9\\ 7y-4x&=&4x-4x+9\\ 7y-4x&=&9 \end{array}\end{align*}$
::7y=4x+97y-4x=4x-4x-4x+97y-4x=9

Next, rewrite the equation so that its order matches $\begin{align*}Ax+By=C\end{align*}$ .
::下一位重写方程式, 使其顺序与 Ax+By=C 匹配。

\begin{aligned} - 4 x + 7 y = 9 \end{aligned}

$\begin{align*}-4x+7y=9\end{align*}$
::- 4x+7y=9

Next, multiply both sides of the equation by -1 to change the coefficient of the ‘ $\begin{align*}x\end{align*}$ ’ variable to a positive value. Simplify the equation.
::接下来,将方程的两边乘以 - 1, 将 ' x ' 变量的系数改为正值。简化方程。

\begin{aligned} \begin{array}{rcl} - 4 x + 7 y & = & 9 \\ - 1 (- 4 x + 7 y) & = & - 1 (9) \\ 4 x - 7 y & = & - 9 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} -4x+7y&=&9\\ -1(-4x+7y) &=& -1(9)\\ 4x-7y&=&-9 \end{array}\end{align*}$
::-4x+7y=9-1(- 4x+7y) @1( 9)4x-7y=9-1(- 4x+7y) @9( 9)4x-7y=9

The answer is $\begin{align*}4x-7y=-9\end{align*}$ .
::答案是 4x-7y9。

Example 3
::例3

Write the following equation in function form.
::以函数形式写入以下方程式。

\begin{aligned} 4 x - 2 y = 16 \end{aligned}

$\begin{align*}4x-2y=16\end{align*}$
::4x-2yy=16

Remember for the equation to be expressed in function form it must be solved for the variable ‘ $\begin{align*}y\end{align*}$ .’
::记住要用函数形式表示方程式,变量“y”必须解决方程式。

First, subtract $\begin{align*}4x\end{align*}$ from both sides of the equation and simplify.
::首先,从方程两侧减去4x并简化。

\begin{aligned} \begin{array}{rcl} 4 x - 2 y & = & 16 \\ 4 x - 4 x - 2 y & = & 16 - 4 x \\ - 2 y & = & - 4 x + 16 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 4x-2y &=& 16\\ 4x-4x-2y &=& 16-4x\\ -2y &=& -4x+16 \end{array}\end{align*}$
::4x-2y=164x-4x-2y=16-4x-2y=16-4x-2y=4x=4x+16

Next, divide each term of the equation by -2 to express the equation in function form.
::下一步,将方程的每个条件除以-2,以函数形式表示方程。

\begin{aligned} \begin{array}{rcl} - 2 y & = & - 4 x + 16 \\ \frac{\overset{1}{- 2 y}}{- 2} & = & \frac{\overset{2}{- 4 x}}{- 2} + \frac{\overset{- 8}{16}}{- 2} \\ y & = & 2 x - 8 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} -2y&=&-4x+16\\ \frac{\overset{1}{\cancel{-2}y}}{\cancel{-2}}&=&\frac{\overset{2}{\cancel{-4}x}}{\cancel{-2}}+\frac{\overset{-8}{\cancel{16}}}{\cancel{-2}}\\ y&=&2x-8 \end{array}\end{align*}$
::-2y4x+16-2y1-24x2-2+16-8-2y=2x-8

The answer is $\begin{align*}y=2x-8\end{align*}$ .
::答案是y=2x-8。

Example 4
::例4

Write the following equation in standard form.
::以标准形式写下以下方程。

\begin{aligned} y + 8 = 6 x \end{aligned}

$\begin{align*}y+8=6x\end{align*}$
::y+8=6x y+8=6x

First, write the standard form of an equation.
::首先,写一个方程的标准格式。

\begin{aligned} A x + B y = C \end{aligned}

$\begin{align*}Ax+By=C\end{align*}$
::Ax+By=C 轴+By=C

Next, subtract 8 from both sides of the equation and simplify the equation.
::接下来,从方程两侧减去8,简化方程。

\begin{aligned} \begin{array}{rcl} y + 8 & = & 6 x \\ y + 8 - 8 & = & 6 x - 8 \\ y & = & 6 x - 8 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} y+8&=&6x\\ y+8-8&=&6x-8\\ y&=&6x-8 \end{array}\end{align*}$
::y+8=6xy+8-8=6x-8y=6x-8

Next, subtract $\begin{align*}6x\end{align*}$ from both sides of the equation and simplify the equation.
::接下来,从方程两侧减去6x,简化方程。

\begin{aligned} \begin{array}{rcl} y & = & 6 x - 8 \\ y - 6 x & = & 6 x - 6 x - 8 \\ y - 6 x & = & - 8 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} y&=&6x-8\\ y-6x&=&6x-6x-8\\ y-6x&=&-8 \end{array}\end{align*}$
::y= 6x- 8y- 6x= 6x- 6x- 8y- 6x- 6x=8

Next, rewrite the equation so that its order matches $\begin{align*}Ax+By=C\end{align*}$ .
::下一位重写方程式, 使其顺序与 Ax+By=C 匹配。

\begin{aligned} - 6 x + y = - 8 \end{aligned}

$\begin{align*}-6x+y=-8\end{align*}$
::- 6x+y+y8

\begin{aligned} \begin{array}{rcl} - 6 x + y & = & - 8 \\ - 1 (- 6 x + y) & = & - 1 (- 8) \\ 6 x - y & = & 8 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} -6x+y&=&-8\\ -1(-6x+y)&=&-1(-8)\\ 6x-y&=&8 \end{array}\end{align*}$
::-6x+y8-1(-6x+y)1(-8)6x-y=8

The answer is $\begin{align*}6x-y=8\end{align*}$ .
::答案是 6x-y=8。

Example 5
::例5

Write the following equation in function form and then use the equation to complete the given table.
::以函数形式写下以下方程式,然后使用方程式填写给定表格。

\begin{aligned} 4 x - y - 1 = 0 \end{aligned}

$\begin{align*}4x-y-1=0\end{align*}$
::4 - y- 1=0

$\begin{align}x\end{align}$	$\begin{align}y\end{align}$
0	-1
2	7
4	15
6	23

First, subtract $\begin{align*}4x\end{align*}$ from both sides of the equation and simplify.
::首先,从方程两侧减去4x并简化。

\begin{aligned} \begin{array}{rcl} 4 x - y - 1 & = & 0 \\ 4 x - 4 x - y - 1 & = & 0 - 4 x \\ - y - 1 & = & - 4 x \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} 4x-y-1&=&0\\ 4x-4x-y-1&=&0-4x\\ -y-1&=&-4x \end{array}\end{align*}$
::4 - y - 1 = 04x - 4x - 4y - 1 = 0 - 4x - y - 1 = 4x - 1 = 4x

Next, subtract add 1 to both sides of the equation and simplify.
::其次,在方程的两侧加上1,并简化。

\begin{aligned} \begin{array}{rcl} - y - 1 & = & - 4 x \\ - y - 1 + 1 & = & - 4 x + 1 \\ - y & = & - 4 x + 1 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} -y-1&=&-4x\\ -y-1+1&=&-4x+1\\ -y&=&-4x+1 \end{array}\end{align*}$
::-y -1 -14x -y -1+1+14x+1 -y4x+1

Next, divide each term of the equation by -1 to express the equation in function form.
::下一步,将方程的每个条件除以-1,以函数形式表示方程。

\begin{aligned} \begin{array}{rcl} - y & = & - 4 x + 1 \\ \frac{\overset{1}{- y}}{- 1} & = & \frac{\overset{4}{- 4} x}{- 1} + \frac{\overset{- 1}{1}}{- 1} \\ y & = & 4 x - 1 \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} -y&=&-4x+1\\ \frac{\overset{1}{\cancel{-}y}}{\cancel{-1}}&=&\frac{\overset{4}{\cancel{-4}}x}{\cancel{-1}}+\frac{\overset{-1}{\cancel{1}}}{\cancel{-1}}\\ y&=&4x-1 \end{array}\end{align*}$
::-y4x+1-y1-1-144x-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1y=4x-1

The answer is $\begin{align*}y=4x-1\end{align*}$ .
::答案是y=4x-1。

Next, use the function form of the equation to calculate the output numbers.
::下一步,使用方程式的函数形式来计算输出数字。

\begin{aligned} \begin{array}{rcl} x & = & 0 \\ y & = & 4 x - 1 \\ y & = & 4 (0) - 1 Substitute x = 0 into the equation . \\ y & = & 0 - 1 Perform the multiplication to clear the parenthesis . \\ y & = & - 1 Simplify the right side of the equation . \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&0\\ y&=&4x-1\\ y&=&4(0)-1 \qquad \text{Substitute } x=0 \text{ into the equation}.\\ y&=&0-1 \qquad \quad \ \text{Perform the multiplication to clear the parenthesis}.\\ y&=&-1 \qquad \qquad \ \text{Simplify the right side of the equation}. \end{array}\end{align*}$
::x=0y=4x-1-y=4(0)-1-Substitut x=0 进入方程式。y=0-1 执行乘法以清除括号。y=1 简化方程式的右侧。

\begin{aligned} \begin{array}{rcl} x & = & 2 \\ y & = & 4 x - 1 \\ y & = & 4 (2) - 1 Substitute x = 2 into the equation . \\ y & = & 8 - 1 Perform the multiplication to clear the parenthesis . \\ y & = & 7 Simplify the right side of the equation . \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&2\\ y&=&4x-1\\ y&=&4(2)-1 \qquad \text{Substitute } x=2 \text{ into the equation}.\\ y&=&8-1 \qquad \quad \ \text{Perform the multiplication to clear the parenthesis}.\\ y&=&7 \qquad \qquad \quad \text{Simplify the right side of the equation}. \end{array}\end{align*}$
::x=2y=4x-1y=4(2)-1Substitut x=2 进入方程式。y=8-1 执行乘法清除括号。y=7 简化方程式的右侧。

\begin{aligned} \begin{array}{rcl} x & = & 4 \\ y & = & 4 x - 1 \\ y & = & 4 (4) - 1 Substitute x = 4 into the equation . \\ y & = & 16 - 1 Perform the multiplication to clear the parenthesis . \\ y & = & 15 Simplify the right side of the equation . \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&4\\ y&=&4x-1\\ y&=&4(4)-1 \qquad \text{Substitute } x=4 \text{ into the equation}.\\ y&=&16-1 \qquad \ \ \ \text{Perform the multiplication to clear the parenthesis}.\\ y&=&15 \qquad \qquad \ \ \text{Simplify the right side of the equation}. \end{array}\end{align*}$
::x=4y=4x-1y=4(4)-1Substitut x=4 进入方程式。y=16-1 执行乘法清除括号。y=15 简化方程式的右侧。

\begin{aligned} \begin{array}{rcl} x & = & 6 \\ y & = & 4 x - 1 \\ y & = & 4 (6) - 1 Substitute x = 4 into the equation . \\ y & = & 24 - 1 Perform the multiplication to clear the parenthesis . \\ y & = & 23 Simplify the right side of the equation . \end{array} \end{aligned}

$\begin{align*}\begin{array}{rcl} x&=&6\\ y&=&4x-1\\ y&=&4(6)-1 \qquad \text{Substitute } x=4 \text{ into the equation}.\\ y&=&24-1 \qquad \ \ \ \text{Perform the multiplication to clear the parenthesis}.\\ y&=&23 \qquad \qquad \ \ \text{Simplify the right side of the equation}. \end{array}\end{align*}$
::x=6y=4x-1y=4(6)-1Substitut x=4 进入方程式。y=24-1 执行乘法清除括号。y=23 简化方程式的右侧。

Then, complete the table by entering the calculated values in the table.
::然后,填写表格时输入表格中的计算值。

Review
::回顾

Find 4 solutions to the function $\begin{align*}3x + y = 24\end{align*}$ . Write your answers as ordered pairs.
::查找 3x+y=24 函数的 4 个解决方案。将您的答案以定购配对形式写入。

Find 4 solutions to the function $\begin{align*}2x-y = 9\end{align*}$ . Write your answers are ordered pairs.
::查找函数 2x-y=9. 的 4 个解决方案。写入您的答案是按顺序排列的对。

Write each equation in standard form.
::以标准格式写下每个方程。

9. $\begin{align*}y = 2x + 3\end{align*}$
::9.y=2x+3 y=2x+3

10. $\begin{align*}y = -4x + 6\end{align*}$
::10. y4x+6

11. $\begin{align*}y = -2x- 4\end{align*}$
::11. y2x-4

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

13. $\begin{align*}y = -3x- 2\end{align*}$
::13. y3x-2

14. $\begin{align*}y = -4x- 6\end{align*}$
::14. y4x-6

15. $\begin{align*}y = 6x- 1\end{align*}$
::15.y=6x-1

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。

Section outline

General

找到两个变量中等式的解决方案

Equations ::平数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)