/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Solve by graphing. $$ \begin... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 63

Solve by graphing. $$ \begin{aligned} 5 x+2 y &=20 \\ -3 x-4 y &=-12 \end{aligned} $$

Short Answer

Expert verified
The solution is where the lines intersect: (4, -2).

Step by step solution

01

鈥� Rewrite Equations in Slope-Intercept Form

To graph the equations, first express them in the slope-intercept form, which is \(y = mx + b\). Start with the first equation: \[5x + 2y = 20\]Solving for y:\[2y = -5x + 20\]\[y = -\frac{5}{2}x + 10\]Next, take the second equation: \[-3x - 4y = -12\]Solving for y:\[-4y = 3x - 12\]\[y = -\frac{3}{4}x + 3\]
02

鈥� Plot the Y-Intercepts

For the first equation \(y = -\frac{5}{2}x + 10\), plot the y-intercept (0, 10). For the second equation \(y = -\frac{3}{4}x + 3\), plot the y-intercept (0, 3).
03

鈥� Use the Slopes to Plot Another Point

For the first equation, the slope is \(-\frac{5}{2}\). This means for each 2 units moved to the right, move 5 units down. Plot another point using this slope. For the second equation, the slope is \(-\frac{3}{4}\). This means for each 4 units moved to the right, move 3 units down. Plot another point using this slope.
04

鈥� Draw the Lines

Draw straight lines through the points plotted for each equation. Extend the lines until they intersect.
05

鈥� Identify the Intersection Point

The solution to the system of equations is the point where the lines intersect. Locate this point on the graph.
06

鈥� Verify the Solution

Verify that the coordinates of the intersection satisfy both original equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
The slope-intercept form of a linear equation is a way to express the equation of a line. The general form is: \( y = mx + b \) Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept. Converting to slope-intercept form helps us easily identify the slope and y-intercept for graphing.
For example, starting with the equations from the problem:
  • First equation: \(5x + 2y = 20\)
  • Second equation: \(-3x - 4y = -12\)
We need to solve for \( y \) in each. For the first equation, rearrange terms and solve:
  • \(2y = -5x + 20\)
  • \(y = -\frac{5}{2} x + 10\)
For the second equation, do the same:
  • \(-4y = 3x - 12\)
  • \(y = -\frac{3}{4} x + 3\)
plotting y-intercepts
Once the equations are in slope-intercept form, we can easily find the y-intercepts. The y-intercept is the value of \( y \) when \( x = 0 \). In the slope-intercept form \( y = mx + b \), the y-intercept is simply \( b \). From the slope-intercept forms:
  • First equation: \( y = -\frac{5}{2} x + 10 \)
  • Second equation: \( y = -\frac{3}{4} x + 3 \)
You can see that the y-intercepts are (0, 10) and (0, 3), respectively. Plot these points on the graph:
  • (0, 10) for the first equation.
  • (0, 3) for the second equation.
Plotting the y-intercept provides a starting point for drawing the line.
graphing linear equations
To graph linear equations, we use the slope and y-intercept. Once the y-intercept is plotted, we use the slope to determine another point.
For each slope:
  • \( y = -\frac{5}{2} x + 10 \): Slope is \(-\frac{5}{2}\). This means for every 2 units we move right on the x-axis, we move 5 units down on the y-axis.
  • \( y = -\frac{3}{4} x + 3 \): Slope is \(-\frac{3}{4}\). This means for every 4 units we move right on the x-axis, we move 3 units down on the y-axis.
From the y-intercept, use the slope to plot another point, then draw a straight line through these points. Repeat for both equations.
Let's add these points to our graph:
  • For \( y = -\frac{5}{2} x + 10 \): Start at (0,10), next point is (2, 5).
  • For \( y = -\frac{3}{4} x + 3 \): Start at (0,3), next point is (4, 0).
intersection of lines
The intersection of two lines on a graph represents the solution to a system of linear equations. It is the point where both equations meet, meaning it satisfies both equations simultaneously.
After graphing the lines:
  • The first line \( y = -\frac{5}{2} x + 10 \) passes through points (0, 10) and (2, 5).
  • The second line \( y = -\frac{3}{4} x + 3 \) passes through points (0, 3) and (4, 0).
Extend these lines on the graph until they intersect. The intersection point is the solution. In this case, the lines intersect at (4, -2).
Verify the solution by plugging these coordinates into both original equations to ensure they hold true:
  • \(5(4) + 2(-2) = 20\)
  • \(-3(4) - 4(-2) = -12\)

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