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Chapter 4: Problem 17

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

Short Answer

Expert verified
No solution; the lines are parallel.

Step by step solution

01

Rewrite equations in slope-intercept form

First, solve each equation for y to get them into the form y = mx + b.For the first equation: \[2y - 2x = 2\] Add 2x to both sides:\[2y = 2x + 2\] Then, divide by 2:\[y = x + 1\]For the second equation:\[2y - 2x = 6\]Add 2x to both sides:\[2y = 2x + 6\]Then, divide by 2:\[y = x + 3\]
02

Graph the first equation

Graph the line given by y = x + 1. This line has a slope of 1 and a y-intercept of 1. Plot the y-intercept (0, 1) and use the slope to find another point (1, 2). Draw the line through these points.
03

Graph the second equation

Graph the line given by y = x + 3. This line has a slope of 1 and a y-intercept of 3. Plot the y-intercept (0, 3) and use the slope to find another point (1, 4). Draw the line through these points.
04

Determine the solution by observing the graph

Observe the graph of both lines. Notice that the lines are parallel because they have the same slope but different y-intercepts. Since parallel lines never intersect, there is no solution to this system of 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 one of the most common forms used, especially for graphing. It follows the formula \(y = mx + b\).
Here, *m* represents the slope of the line, and *b* is the y-intercept.
The y-intercept is the point where the line crosses the y-axis.
For example, in the equation \(y = x + 1\), the slope (m) is 1, and the y-intercept (b) is 1.
This tells us the line rises at a rate of 1 unit up for every 1 unit it moves to the right.
The slope and y-intercept help us quickly graph a line by plotting the y-intercept and using the slope to find another point.
parallel lines
Parallel lines are lines in the same plane that never intersect, no matter how far they extend.
One key characteristic of parallel lines is that they have the same slope.
For example, in the system of equations given, \(y = x + 1\) and \(y = x + 3\) both have a slope of 1.
Because their slopes are identical, these lines will never cross each other.
The only difference between these parallel lines is their y-intercepts.
In this case, the first line crosses the y-axis at (0,1) and the second line at (0,3).
It's useful to remember that parallel lines indicate that the system of equations has no points in common.
no solution
When graphing a system of equations, if the lines are parallel and do not intersect, the system has no solution.
This means there are no values of x and y that satisfy both equations simultaneously.
In the given example, \(y = x + 1\) and \(y = x + 3\) represent two parallel lines with the same slope (1) but different y-intercepts (1 and 3, respectively).
Since they never meet, there are no common points that solve both equations.
This situation is also described as the system being 'inconsistent'.
Knowing how to identify this graphically helps us quickly determine if a system of equations has no solution just by looking at the slope and intercepts.

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Most popular questions from this chapter

Solve each system using the Gauss-Jordan elimination method. $$ \begin{aligned} 4 x-2 y+2 z &=2 \\ 2 x-y+z &=1 \\ -2 x+y-z &=-1 \end{aligned} $$

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Determine the row operation that was used to convert each given augmented matrix into the equivalent augmented matrix that follows it. $$ \left[\begin{array}{rr|r} 1 & -1 & -1 \\ 0 & 5 & 15 \end{array}\right],\left[\begin{array}{rr|r} 1 & -1 & -1 \\ 0 & 1 & 3 \end{array}\right] $$

Solve each system using the Gauss-Jordan elimination method. $$ \begin{aligned} x+y &=3 \\ -3 x+y &=-1 \end{aligned} $$
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