### Review Exercises

## Rectangular Coordinate System

**Plot Points in a Rectangular Coordinate System**

In the following exercises, plot each point in a rectangular coordinate system.

- ⓐ $\left(\mathrm{-1},\mathrm{-5}\right)$
- ⓑ $\left(\mathrm{-3},4\right)$
- ⓒ $\left(2,\mathrm{-3}\right)$
- ⓓ $\left(1,\frac{5}{2}\right)$

- ⓐ $\left(4,3\right)$
- ⓑ $\left(\mathrm{-4},3\right)$
- ⓒ $\left(\mathrm{-4},\mathrm{-3}\right)$
- ⓓ $\left(4,\mathrm{-3}\right)$

- ⓐ $\left(\mathrm{-2},0\right)$
- ⓑ $\left(0,\mathrm{-4}\right)$
- ⓒ $\left(0,5\right)$
- ⓓ $\left(3,0\right)$

- ⓐ $\left(2,\frac{3}{2}\right)$
- ⓑ $\left(3,\frac{4}{3}\right)$
- ⓒ $\left(\frac{1}{3},\mathrm{-4}\right)$
- ⓓ $\left(\frac{1}{2},\mathrm{-5}\right)$

**Identify Points on a Graph**

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

**Verify Solutions to an Equation in Two Variables**

In the following exercises, which ordered pairs are solutions to the given equations?

$5x+y=10$

- ⓐ $\left(5,1\right)$
- ⓑ $\left(2,0\right)$
- ⓒ $\left(4,\mathrm{-10}\right)$

**Complete a Table of Solutions to a Linear Equation in Two Variables**

In the following exercises, complete the table to find solutions to each linear equation.

$y=4x-1$

$x$ |
$y$ |
$(x,y)$ |

0 | ||

1 | ||

$\mathrm{-2}$ |

$x+2y=5$

$x$ |
$y$ |
$(x,y)$ |

0 | ||

1 | ||

$\mathrm{-1}$ |

**Find Solutions to a Linear Equation in Two Variables**

In the following exercises, find three solutions to each linear equation.

$x+y=3$

$y=3x+1$

## Graphing Linear Equations

**Recognize the Relation Between the Solutions of an Equation and its Graph**

In the following exercises, for each ordered pair, decide:

- ⓐ Is the ordered pair a solution to the equation?
- ⓑ Is the point on the line?

$y=\text{\u2212}x+4$

ⓐ $\left(0,4\right)$

ⓑ $\left(\mathrm{-1},3\right)$

ⓒ $\left(2,2\right)$

ⓓ $\left(\mathrm{-2},6\right)$

$y=\frac{2}{3}x-1$

ⓐ $\left(0,\mathrm{-1}\right)$

ⓑ $(3,1)$

ⓒ $(\mathrm{-3},\mathrm{-3})$

ⓓ $(6,4)$

**Graph a Linear Equation by Plotting Points**

In the following exercises, graph by plotting points.

$y=4x-3$

$y=\frac{1}{2}x+3$

$2x+y=7$

**Graph Vertical and Horizontal lines**

In the following exercises, graph each equation.

$y=\mathrm{-2}$

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

$y=\mathrm{-2}x$ and $y=\mathrm{-2}$

## Graphing with Intercepts

**Identify the x- and y-Intercepts on a Graph**

In the following exercises, find the *x*- and *y*-intercepts.

**Find the x- and y-Intercepts from an Equation of a Line**

In the following exercises, find the intercepts of each equation.

$x+y=5$

$x+2y=6$

$y=\frac{3}{4}x-12$

**Graph a Line Using the Intercepts**

In the following exercises, graph using the intercepts.

$\text{\u2212}x+3y=3$

$x-y=4$

$2x-4y=8$

## Slope of a Line

**Use Geoboards to Model Slope**

In the following exercises, find the slope modeled on each geoboard.

In the following exercises, model each slope. Draw a picture to show your results.

$\frac{1}{3}$

$-\frac{2}{3}$

**Use $m=\frac{\text{rise}}{\text{run}}$ to find the Slope of a Line from its Graph**

In the following exercises, find the slope of each line shown.

**Find the Slope of Horizontal and Vertical Lines**

In the following exercises, find the slope of each line.

$y=2$

$x=\mathrm{-3}$

**Use the Slope Formula to find the Slope of a Line between Two Points**

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

$\left(\mathrm{-1},\mathrm{-1}\right),\left(0,5\right)$

$\left(\mathrm{-5},\mathrm{-2}\right),\left(3,2\right)$

**Graph a Line Given a Point and the Slope**

*In the following exercises, graph each line with the given point and slope.*

$\left(2,\mathrm{-2}\right)$; $m=\frac{5}{2}$

*x*-intercept $\mathrm{-4}$; $m=3$

**Solve Slope Applications**

In the following exercises, solve these slope applications.

The roof pictured below has a rise of 10 feet and a run of 15 feet. What is its slope?

## Intercept Form of an Equation of a Line

**Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line**

In the following exercises, use the graph to find the slope and *y*-intercept of each line. Compare the values to the equation $y=mx+b$.

$y=4x-1$

**Identify the Slope and y-Intercept from an Equation of a Line**

In the following exercises, identify the slope and *y*-intercept of each line.

$y=\mathrm{-4}x+9$

$5x+y=10$

**Graph a Line Using Its Slope and Intercept**

In the following exercises, graph the line of each equation using its slope and *y*-intercept.

$y=2x+3$

$y=-\frac{2}{5}x+3$

In the following exercises, determine the most convenient method to graph each line.

$x=5$

$2x+y=5$

$y=x+2$

**Graph and Interpret Applications of Slope–Intercept**

Katherine is a private chef. The equation $C=6.5m+42$ models the relation between her weekly cost, *C*, in dollars and the number of meals, *m*, that she serves.

- ⓐ Find Katherine’s cost for a week when she serves no meals.
- ⓑ Find the cost for a week when she serves 14 meals.
- ⓒ Interpret the slope and
*C*-intercept of the equation. - ⓓ Graph the equation.

Marjorie teaches piano. The equation $P=35s-250$ models the relation between her weekly profit, *P*, in dollars and the number of student lessons, *s*, that she teaches.

- ⓐ Find Marjorie’s profit for a week when she teaches no student lessons.
- ⓑ Find the profit for a week when she teaches 20 student lessons.
- ⓒ Interpret the slope and
*P*–intercept of the equation. - ⓓ Graph the equation.

**Use Slopes to Identify Parallel Lines**

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

$4x-3y=\mathrm{-1};\phantom{\rule{0.2em}{0ex}}y=\frac{4}{3}x-3$

**Use Slopes to Identify Perpendicular Lines**

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

$y=5x-1;10x+2y=0$

## Find the Equation of a Line

**Find an Equation of the Line Given the Slope and y-Intercept**

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope $\frac{1}{3}$ and $y\text{-intercept}$ $\left(0,\mathrm{-6}\right)$

slope $0$ and $y\text{-intercept}$ $\left(0,4\right)$

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

**Find an Equation of the Line Given the Slope and a Point**

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

$m=-\frac{1}{4}$, point $\left(\mathrm{-8},3\right)$

Horizontal line containing $\left(\mathrm{-2},7\right)$

**Find an Equation of the Line Given Two Points**

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

$\left(2,10\right)$ and $\left(\mathrm{-2},\mathrm{-2}\right)$

$\left(3,8\right)$ and $\left(3,\mathrm{-4}\right)$.

**Find an Equation of a Line Parallel to a Given Line**

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

line $y=\mathrm{-3}x+6$, point $\left(1,\mathrm{-5}\right)$

line $x=4$, point $\left(\mathrm{-2},\mathrm{-1}\right)$

**Find an Equation of a Line Perpendicular to a Given Line**

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line $y=-\frac{4}{5}x+2$, point $\left(8,9\right)$

line $y=3$, point $\left(\mathrm{-1},\mathrm{-3}\right)$

## Graph Linear Inequalities

**Verify Solutions to an Inequality in Two Variables**

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality $y<x-3$:

- ⓐ $\left(0,1\right)$
- ⓑ $\left(\mathrm{-2},\mathrm{-4}\right)$
- ⓒ $\left(5,2\right)$
- ⓓ $\left(3,\mathrm{-1}\right)$
- ⓔ $\left(\mathrm{-1},\mathrm{-5}\right)$

Determine whether each ordered pair is a solution to the inequality $x+y>4$:

- ⓐ $\left(6,1\right)$
- ⓑ $\left(\mathrm{-3},6\right)$
- ⓒ $\left(3,2\right)$
- ⓓ $\left(\mathrm{-5},10\right)$
- ⓔ $\left(0,0\right)$

**Recognize the Relation Between the Solutions of an Inequality and its Graph**

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line $y=\text{\u2212}x+2$.

Write the inequality shown by the shaded region in the graph with the boundary line $x+y=\mathrm{-4}$.

**Graph Linear Inequalities**

In the following exercises, graph each linear inequality.

Graph the linear inequality $y>\frac{2}{5}x-4$.

Graph the linear inequality $x-y\le 5$.

Graph the linear inequality $y\le \mathrm{-3}x$.