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Chapter 3: Problem 31

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(-x+y=-1\)

Short Answer

Expert verified
(0, -1), (1, 0), (2, 1), and plot these points.

Step by step solution

01

- Rewrite the Equation in Slope-Intercept Form

Rewrite the given equation \(-x + y = -1\) in the form \(y = mx + b\). Add \(-x\) to both sides to get \(y = x - 1\).
02

- Find the y-intercept

To find the y-intercept, let \(x = 0\) and solve for \(y\). Substituting \(x = 0\) into the equation \(y = x - 1\), the y-intercept is: \(y = 0 - 1 = -1\). So, the y-intercept is \((0, -1)\).
03

- Find the x-intercept

To find the x-intercept, let \(y = 0\) and solve for \(x\). Substituting \(y = 0\) into the equation \(-x + y = -1\), we get \(-x + 0 = -1\), solving for \(x\): \(-x = -1\) which gives \(x = 1\). Thus, the x-intercept is \((1, 0)\).
04

- Find a Third Solution

To find a third solution, choose another value for \(x\) and solve for \(y\). For example, let \(x = 2\). Substituting \(x = 2\) into \(y = x - 1\), we get \(y = 2 - 1 = 1\). So, the third solution is \((2, 1)\).
05

- Graph the Equation

Plot the points found: the y-intercept \((0, -1)\), the x-intercept \((1, 0)\), and the third solution \((2, 1)\). Draw a straight line through these points to graph the equation \(-x + y = -1\).

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Key Concepts

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

y-intercept
In a linear equation, the *y-intercept* is the point where the line crosses the y-axis. This means the value of y when x is zero. To find the y-intercept from an equation like \[ -x + y = -1 \], set x to 0 and solve for y.

From the step-by-step solution, we know that when x = 0, the equation becomes: \[ y = 0 - 1 = -1 \].
The y-intercept here is (0, -1).
This point is crucial when graphing the line because it helps to define where the line starts on the y-axis.
x-intercept
The *x-intercept* is similar to the y-intercept but for the x-axis. It is the point where the line crosses the x-axis, meaning the value of x when y is zero. To find the x-intercept, set y to 0 and solve for x.

For the equation \[ -x + y = -1 \], when y = 0, the equation simplifies to \[ -x + 0 = -1 \] which leads to \[ -x = -1 \text{ or } x = 1 \]. Therefore, the x-intercept is (1, 0).
This point is important for graphing the equation as it helps to define where the line crosses the x-axis.
graphing linear equations
Graphing linear equations involves plotting points and drawing a straight line through them. The main points needed are the intercepts and any additional solutions. For the equation \[ -x + y = -1 \], we already have:
  • y-intercept (0, -1)
  • x-intercept (1, 0)

Another point can be found by choosing any x-value. In the step-by-step solution, we used x = 2, and found y = 1, giving the point (2, 1).
Plot these points on a graph:
  • (0, -1)
  • (1, 0)
  • (2, 1)

Draw a straight line through these points, and you have a graph of the equation.
slope-intercept form
The *slope-intercept form* of a linear equation is \[ y = mx + b \], where m is the slope and b is the y-intercept. To convert \[ -x + y = -1 \] to slope-intercept form, solve for y:
\[ y = x - 1 \]
Here, m, the slope, is 1 and b, the y-intercept, is -1.
The slope indicates how steep the line is and in which direction it goes. A positive slope means the line rises as it goes from left to right, and a negative slope means it falls. In this case, the slope is 1, meaning the line rises evenly.
finding solutions
To find *solutions* to a linear equation, choose a value for x and solve for y, or vice versa. These solutions are coordinates that lie on the line.
The step-by-step solution used x = 2 to find another point:
  • When x = 2, \[ y = 2 - 1 = 1 \], so the solution is (2, 1).

You can choose any value for x to find more points. This method helps in thoroughly plotting the line on a graph by providing as many points as needed to ensure accuracy.
Keep in mind that finding solutions helps not only in graphing but also in understanding the relationship between x and y in the equation.

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Most popular questions from this chapter

(a) find three solutions of the equation. (b) graph the equation. \(y=-x\)

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use the slope formula to find the slope of the line that passes through \((6,2)\) and \((6,7)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{6-6}{7-2} \\ m &=\frac{0}{5} \\ m &=0 \end{aligned} $$

Use the slope formula to find the slope of the line that passes through the points. \((-1,5) ;(6,13)\)

For exercises 89-92, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the slope of the line that passes through \((7,1)\) and \((9,4)\). Incorrect Answer: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) $$ \begin{aligned} m &=\frac{9-7}{4-1} \\ m &=\frac{2}{3} \end{aligned} $$

Use the slope formula to find the slope of the line that passes through the points. \((-30,55) ;(-5,80)\)
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