### Key Concepts

## 4.1 Use the Rectangular Coordinate System

**Sign Patterns of the Quadrants**

$\begin{array}{cccccccccc}\text{Quadrant I}\hfill & & & \text{Quadrant II}\hfill & & & \text{Quadrant III}\hfill & & & \text{Quadrant IV}\hfill \\ (x,y)\hfill & & & (x,y)\hfill & & & (x,y)\hfill & & & (x,y)\hfill \\ (+,+)\hfill & & & (\text{\u2212},+)\hfill & & & (\text{\u2212},\text{\u2212})\hfill & & & (+,\text{\u2212})\hfill \end{array}$**Points on the Axes**

- On the
*x*-axis, $y=0$. Points with a*y*-coordinate equal to 0 are on the*x*-axis, and have coordinates $(a,0)$. - On the
*y*-axis, $x=0$. Points with an*x*-coordinate equal to 0 are on the*y*-axis, and have coordinates $(0,b).$

- On the
**Solution of a Linear Equation**

- An ordered pair $\left(x,y\right)$ is a solution of the linear equation $Ax+By=C$, if the equation is a true statement when the
*x*- and*y*- values of the ordered pair are substituted into the equation.

- An ordered pair $\left(x,y\right)$ is a solution of the linear equation $Ax+By=C$, if the equation is a true statement when the

## 4.2 Graph Linear Equations in Two Variables

**Graph a Linear Equation by Plotting Points**- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
- Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

## 4.3 Graph with Intercepts

**Find the***x*- and*y*- Intercepts from the Equation of a Line- Use the equation of the line to find the
*x*- intercept of the line, let $y=0$ and solve for*x*. - Use the equation of the line to find the
*y*- intercept of the line, let $x=0$ and solve for*y*.

- Use the equation of the line to find the
**Graph a Linear Equation using the Intercepts**- Step 1.
Find the
*x*- and*y*- intercepts of the line.

Let $y=0$ and solve for*x*.

Let $x=0$ and solve for*y*. - Step 2. Find a third solution to the equation.
- Step 3. Plot the three points and then check that they line up.
- Step 4. Draw the line.

- Step 1.
Find the
**Strategy for Choosing the Most Convenient Method to Graph a Line:**- Consider the form of the equation.
- If it only has one variable, it is a vertical or horizontal line.

$x=a$ is a vertical line passing through the*x*- axis at $a$

$y=b$ is a horizontal line passing through the*y*- axis at $b$. - If
*y*is isolated on one side of the equation, graph by plotting points. - Choose any three values for
*x*and then solve for the corresponding*y*- values. - If the equation is of the form $ax+by=c$, find the intercepts. Find the
*x*- and*y*- intercepts and then a third point.

## 4.4 Understand Slope of a Line

**Find the Slope of a Line from its Graph using**$m=\frac{\text{rise}}{\text{run}}$- Step 1. Locate two points on the line whose coordinates are integers.
- Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
- Step 3. Count the rise and the run on the legs of the triangle.
- Step 4. Take the ratio of rise to run to find the slope.

**Graph a Line Given a Point and the Slope**- Step 1. Plot the given point.
- Step 2. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
- Step 3. Starting at the given point, count out the rise and run to mark the second point.
- Step 4.
Connect the points with a line.

**Slope of a Horizontal Line**- The slope of a horizontal line, $y=b$, is 0.

**Slope of a vertical line**- The slope of a vertical line, $x=a$, is undefined

## 4.5 Use the Slope-Intercept Form of an Equation of a Line

- The slope–intercept form of an equation of a line with slope $m$ and
*y*-intercept, $\left(0,b\right)$ is, $y=mx+b$. **Graph a Line Using its Slope and***y*-Intercept- Step 1. Find the slope-intercept form of the equation of the line.
- Step 2.
Identify the slope and
*y*-intercept. - Step 3.
Plot the
*y*-intercept. - Step 4. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
- Step 5.
Starting at the
*y*-intercept, count out the rise and run to mark the second point. - Step 6. Connect the points with a line.

**Strategy for Choosing the Most Convenient Method to Graph a Line:**Consider the form of the equation.- If it only has one variable, it is a vertical or horizontal line.

$x=a$ is a vertical line passing through the*x*-axis at $a$.

$y=b$ is a horizontal line passing through the*y*-axis at $b$. - If $y$ is isolated on one side of the equation, in the form $y=mx+b$, graph by using the slope and
*y*-intercept.

Identify the slope and*y*-intercept and then graph. - If the equation is of the form $Ax+By=C$, find the intercepts.

Find the*x*- and*y*-intercepts, a third point, and then graph.

- If it only has one variable, it is a vertical or horizontal line.
- Parallel lines are lines in the same plane that do not intersect.
- Parallel lines have the same slope and different
*y*-intercepts. - If
*m*_{1}and*m*_{2}are the slopes of two parallel lines then ${m}_{1}={m}_{2}.$ - Parallel vertical lines have different
*x*-intercepts.

- Parallel lines have the same slope and different
- Perpendicular lines are lines in the same plane that form a right angle.
- If ${m}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{m}_{2}$ are the slopes of two perpendicular lines, then ${m}_{1}\xb7{m}_{2}=\mathrm{-1}$ and ${m}_{1}=\frac{\mathrm{-1}}{{m}_{2}}$.
- Vertical lines and horizontal lines are always perpendicular to each other.

## 4.6 Find the Equation of a Line

**To Find an Equation of a Line Given the Slope and a Point**- Step 1. Identify the slope.
- Step 2. Identify the point.
- Step 3. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$.
- Step 4. Write the equation in slope-intercept form.

**To Find an Equation of a Line Given Two Points**- Step 1. Find the slope using the given points.
- Step 2. Choose one point.
- Step 3. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$.
- Step 4. Write the equation in slope-intercept form.

**To Write and Equation of a Line**- If given slope and
*y*-intercept, use slope–intercept form $y=mx+b$. - If given slope and a point, use point–slope form $y-{y}_{1}=m\left(x-{x}_{1}\right)$.
- If given two points, use point–slope form $y-{y}_{1}=m\left(x-{x}_{1}\right)$.

- If given slope and
**To Find an Equation of a Line Parallel to a Given Line**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the parallel line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$.
- Step 5. Write the equation in slope-intercept form.

**To Find an Equation of a Line Perpendicular to a Given Line**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the perpendicular line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$.
- Step 5. Write the equation in slope-intercept form.

## 4.7 Graphs of Linear Inequalities

**To Graph a Linear Inequality**- Step 1.
Identify and graph the boundary line.

If the inequality is $\le \text{or}\ge $, the boundary line is solid.

If the inequality is < or >, the boundary line is dashed. - Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
- Step 3.
Shade in one side of the boundary line.

If the test point is a solution, shade in the side that includes the point.

If the test point is not a solution, shade in the opposite side.

- Step 1.
Identify and graph the boundary line.