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Chapter 8: Problem 19

Solve each system by graphing. $$\begin{aligned} &3 y+2 x=6\\\ &y=-\frac{2}{3} x-1 \end{aligned}$$

Short Answer

Expert verified
The lines are parallel, and there is no solution.

Step by step solution

01

Convert equations to slope-intercept form

First, convert each equation into the slope-intercept form, which is given by \[y = mx + b\], where \( m\) is the slope and \( b\) is the y-intercept. The second equation is already in slope-intercept form \( y = -\frac{2}{3} x - 1 \). To convert the first equation, solve for \( y \): \[3y + 2x = 6\]Subtract \( 2x\) from both sides: \[3y = -2x + 6\]Divide by 3: \[y = -\frac{2}{3} x + 2\]
02

Identify the slope and y-intercept

For the first equation, \( y = -\frac{2}{3} x + 2 \), the slope \( m \) is \( -\frac{2}{3} \) and the y-intercept \( b \) is 2. For the second equation, \( y = -\frac{2}{3} x - 1 \), the slope \( m \) is \( -\frac{2}{3} \) and the y-intercept \( b \) is -1.
03

Graph the equations

Plot each line on a graph using their slopes and y-intercepts: 1. For the first equation \( y = -\frac{2}{3} x + 2 \), start at the y-intercept (0, 2). From this point, use the slope \( -\frac{2}{3} \) to plot the next points. Go down 2 units and right 3 units. 2. For the second equation \( y = -\frac{2}{3} x - 1 \), start at the y-intercept (0, -1). From this point, use the slope \( -\frac{2}{3} \) to plot the next points. Go down 2 units and right 3 units.
04

Analyze the graph

Both lines have the same slope and are therefore parallel. Parallel lines never intersect. As a result, this system of equations has no solution.

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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 wonderful way to understand the basic features of a line. It鈥檚 given by the formula: \[y = mx + b\].

Here, \(m\) stands for the slope which indicates the steepness of the line, and \(b\) is the y-intercept indicating where the line crosses the y-axis.

When you convert any linear equation into this form, you can easily spot the slope and the y-intercept. For example, from your exercise, we had equations: \[3y + 2x = 6 \] and \[y = -\frac{2}{3}x - 1 \].

The second equation is already in slope-intercept form. To convert the first one into this form, follow these steps:
  • Solve for \(y\) by isolating it on one side of the equation.
  • First, subtract \(2x\) from both sides to get: \[3y = -2x + 6 \]
  • Then, divide everything by 3: \[y = -\frac{2}{3}x + 2 \]
Now you have both equations in slope-intercept form!
graphing linear equations
Graphing linear equations is like drawing a roadmap of your equations! Once you know the slope and y-intercept, you can plot the line on a graph.

Let鈥檚 use the equations we converted earlier: \[y = -\frac{2}{3}x + 2 \] and \[y = -\frac{2}{3}x - 1 \].
  • Start by plotting the y-intercept. For the first equation, it鈥檚 (0, 2). For the second, it's (0, -1).
  • From these intercept points, use the slope to find more points. The slope of \(-\frac{2}{3}\) means you go down 2 units and right 3 units from the intercept.
Keep plotting points until you have a clear line, and draw it out. Do this for both equations and you鈥檒l visually see each line on the same graph.
parallel lines and no solution
When you graph linear equations and they end up being parallel, something interesting happens.

For the system \[y = -\frac{2}{3}x + 2 \] and \[y = -\frac{2}{3}x - 1 \], you may notice both lines have the same slope \(-\frac{2}{3}\), but different y-intercepts.

Because their slopes are the same and they are not the same line, these lines are parallel and will never intersect.
  • If the lines don鈥檛 intersect, they don鈥檛 have any points in common.
  • Thus, the system of equations has no solution.
This is a key point in understanding systems of equations: parallel lines do not share any solutions.

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Most popular questions from this chapter

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