9.6 使用拦截
Section outline
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Mary Ellen has $60 to spend at the craft store and she is very interested in buying either scrapbooks that cost $10 or fashion stickers that cost $5 per package. Before making a purchase, she needs to know how many scrapbooks or how many packages of stickers she can buy with her money. How can Mary Ellen figure this out?
::玛丽·艾伦在手工艺品商店有60美元,她非常有兴趣购买10美元的剪贴纸,或者每包5美元的时装贴纸。 在购买之前,她需要知道有多少剪贴纸,或者她可以用钱买多少套贴纸。 玛丽·埃伦怎样才能弄明白这一点呢?In this concept, you will learn to use intercepts .
::在这个概念中,你会学会使用拦截。Intercepts
::拦截Consider the following linear graph.
::考虑下线性图。The above graph models the linear function . The function is written in standard form . Notice that the line passes through the -axis at the point . The point where the graph intersects the horizontal -axis is called the - intercept . The -value of any point on the -axis is zero. The -axis is really the line having the equation .
::上图模式的线性函数 3x+2y=6. 函数以标准格式 Ax+By=C 写成。请注意,该线在点(2,0) 穿过 x 轴。该线通过x 轴,该点将水平 x 轴交叉称为 x 界面。x 轴上任何点的 Y 值为零。x 轴实际上是具有 y=0 方程式的线条。The line also passes through the -axis at the point . The point where the graph intersects the vertical -axis is called the - intercept . The -value of any point on the -axis is zero. The -axis is really the line having the equation .
::线条在点( 0, 3) 也通过 y 轴。 图形交叉垂直 y 轴的点被称为 y 界面。 Y 轴上任何点的 x 值为 0。 Y 轴是具有等式 x=0 的线条 。Remember, only two points are needed to draw a straight line. Therefore , the graph of the linear function was drawn by plotting the - and -intercepts and using a straight edge to join the two plotted points.
::记住, 绘制直线线线条只需要两个点。 因此, 线性函数 3x+2y=6 的图形是用绘制 X 和 y 界面, 并使用直边缘加入两个绘图点来绘制的 。From a given function, the intercepts can be determined using algebra. Let’s look at an example.
::从给定的函数中,拦截量可以用代数来确定。让我们来举一个例子。
::5x-3y=15First, determine the Substitute in the equation.
-intercept.
::首先,确定 X 界面。 方程中的替代 y=0 。
::5x-3y=155x-3(0)=15Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::5x-3(0)=155x-0=155x=155x=15Next, divide both sides of the equation by ‘5’ to solve for ‘ ’.
::接下来,将等式的两边除以 ' 5 ' , 以解答 'x ' 。
::5x=1551x5=1535x=3The answer is 3.
::答案是3The
-intercept is .
::x 拦截是 (3,0) 。Second, determine the
-intercept. Substitute in the equation.
::第二,确定 Y 界面。 方程式中的替代 x=0 。
::5x-3y=155(0)-3y=15Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::5(0)-3y=150-3y=15-3y=15Next, divide both sides of the equation by ‘-3’ to solve for ‘ ’.
::接下来,将方程式两边除以 ' 3 ' ,
::-3y=15-31-3=15-5-3-3y=5The answer is -5.
::答案是 -5The .
-intercept is
::y 界面是 (0,-5) 。Now, the values of the .
- and -intercepts can be used to plot the graph of the linear function
::现在, x 和 y 界面的值可用于绘制线性函数 5x-3y=15 的图。Examples
::实例Example 1
::例1Earlier, you were given a problem about Mary Ellen and the scrapbooks or stickers. She needs to figure out how many scrapbooks or how many packages of fashion stickers she can buy with her $60.00. How can she do this?
::早些时候,有人给了你一个关于Mary Ellen和剪贴纸或贴纸的问题。她需要弄清楚她能用60美元买到多少剪贴纸或多少套时装贴纸。她怎么能这样做呢?Mary Ellen can use the
- and -intercepts of a linear function.
::Mary Ellen可以使用线性函数的 X 和 Y 界面。First, write a linear function to represent the information given.
::首先,写一个线性函数以代表给定的信息。Let ‘ ’ represent the number of scrapbooks that cost $10 each and let ‘ ’ represent the number of packages of stickers that cost $5.00 each. She has $60.00 to spend.
::让`x ' 代表每本10美元的废纸数量,让`y ' 代表每张5美元的成套贴纸数量。 她有60美元可花。The linear function to model the given information is:
::用于模拟给定信息的线性函数为:
::10x+5y=60Next, determine
- and -intercepts of the linear function.
::下一步,确定线性函数的 X 和 Y 界面。First, determine the
-intercept. Substitute in the equation.
::首先,确定 X 界面。 方程中的替代 y=0 。
::10x+5y=6010x+5(0)=60Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::10x+5(0)=6010x+0=6010x=60Next, divide both sides of the equation by ‘10’ to solve for ‘ ’.
::接下来,将等式的两边除以 ' 10 ' , 解决 'x ' 。
::10x=60101x10=60610x=6The answer is 6.
::答案是6个The
-intercept is .
::x 拦截( 6, 0) 。Second, determine the
-intercept. Substitute in the equation.
::第二,确定 Y 界面。 方程式中的替代 x=0 。
::10x+5y=6010(0)+5y=60Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::10(0)+5y=600+5y=60Next, divide both sides of the equation by ‘30’ to solve for ‘ ’.
::接下来,将方程式两边除以 '30 ' ,
::5y=6051y5=60125y=12The answer is 12.
::答案是12岁The
-intercept is .
::y 界面是 (0, 12) 。Mary Ellen can buy either 6 scrapbooks and no stickers or 12 packages of stickers and no scrapbooks.
::Mary Ellen既可以买6本剪贴纸,又不能买贴纸,也可以买12包贴纸,也不能买剪贴纸。Example 2
::例2For the given linear function, determine the
- and -intercepts.
::对于给定的线性函数,确定 x 和 y 界面。
::4-3y24First, determine the
-intercept. Substitute in the equation.
::首先,确定 X 界面。 方程中的替代 y=0 。
::4 - 3y244x-3(0)24Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::4x-3(0)244x-024x24Next, divide both sides of the equation by ‘4’ to solve for ‘ ’.
::接下来,将方程式两边除以 '4 ' ' ' ' ' ' ' ' ' , 解决`x ' 。
::4x 4x2441x4}24-64x#6The answer is -6.
::答案是 -6The
-intercept is .
::X 拦截( - 6,0) 。Second, determine the
-intercept. Substitute in the equation.
::第二,确定 Y 界面。 方程式中的替代 x=0 。
::4x-3y244(0)-3y24Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::4(0)-(3)-(240)-(3)-(24)-(3)-(24)Next, divide both sides of the equation by ‘-3’ to solve for ‘ ’.
::接下来,将方程式两边除以 ' 3 ' ,
::- 3 - 24 - 31 - 3 - 24 - 8 - 8 - 3y=8The answer is 8.
::答案是8岁The
-intercept is .
::y 界面是( 0, 8) 。Example 2
::例2For the given linear function, use the
-and -intercepts to draw the graph:
::对于给定的线性函数,使用 x 和 y 界面绘制图形:
::6x-4y=24First, determine the
-intercept. Substitute in the equation.
::首先,确定 X 界面。 方程中的替代 y=0 。
::6x-4y=246x-44(0)=24Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::6x-4(0)=246x-0=246x=24Next, divide both sides of the equation by ’.
to solve for ‘
::接下来,将方程式两边除以 '6 ' ' ' ' ' ' ' , 解决`x ' 。
::6x=2461x6=2446x=4The answer is 4.
::答案是4。The
-intercept is .
::x 界面为 (4,0) 。Second, determine the
-intercept. Substitute in the equation.
::第二,确定 Y 界面。 方程式中的替代 x=0 。
::6x-4y=246(0)-4y=24Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::6(0)-4y=240-4y=24-4y=24Next, divide both sides of the equation by ‘-4’ to solve for ‘ ’.
::接下来,将方程式两边除以 ' 4 ' ,
::− 4y=24-41y-44=24-6-4y=6-6-4y6The answer is -6.
::答案是 -6The
-intercept is .
::y 界面是 (0, - 6) 。Now, the values of the
- and -intercepts can be used to plot the graph of the linear function.
::现在, x 和 y 界面的值可用于绘制线性函数的图形。
::6x-4y=24Example 3
::例3For the following graph, name the -intercepts.
- and
::下图中,请标明 X 和 y 界面。The graph crosses the
-axis at the point and the -axis at the point .
::图表在点(30,0)横过x轴,在点(0,-50)横过Y轴。The
-intercept is and the -intercept is .
::x 拦截( 30,0) , y 拦截( 0,-50) 。Example 4
::例4For the given linear function, determine the -intercepts.
- and
::对于给定的线性函数,确定 x 和 y 界面。
::15x+30y=120First, determine the
-intercept. Substitute in the equation.
::首先,确定 X 界面。 方程中的替代 y=0 。
::15x+30y=12015x+30(0)=120Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::15x+30(0)=12015x+0=12015x=12015x=120Next, divide both sides of the equation by ‘15’ to solve for ‘ ’.
::接下来,将等式两边除以 ' 15 ' 来解答 'x ' 。
::15x=120151x15=120815x=8The answer is 8.
::答案是8岁The
-intercept is .
::X 界面是 8,0 个 。Second, determine the
-intercept. Substitute in the equation.
::第二,确定 Y 界面。 方程式中的替代 x=0 。
::15x+30y=12015(0)+30y=120Next, perform the multiplication to clear the parenthesis.
::下一步,执行乘法清除括号。
::15(0)+30y=1200+30y=12030y=120Next, divide both sides of the equation by ‘30’ to solve for ‘ ’.
::接下来,将方程式两边除以 '30 ' ,
::30y=120301y30=120430y=4The answer is 4.
::答案是4。The
-intercept is .
::Y 界面是 0, 4 。Review
::回顾Determine the and -intercepts of each equation. There will be two answers for each equation.
::确定每个方程式的 x 和 Y 界面。 每个方程式将有两个答案 。1.
::1. 3x+4y=122.
::2. 6x+2y=123.
::3. 4x+5y=204.
::4. 4x+2y=85.
::5. 3x+5y=156.
::6.-2x+3y=67.
::7.-3x+y=98.
::8.-2x-2y=69.
::9. 7x+3y=2110.
::10. 2x+9y=36Look at each graph and identify the
and -intercept of each equation. Each graph will have two answers.
::查看每个图形, 并识别每个方程式的 x 和 Y 界面。 每个图形将有两个答案 。11.
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15.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 。