Key Concepts

4.1 Use the Rectangular Coordinate System

Sign Patterns of the Quadrants
$\begin{matrix} Quadrant I & Quadrant II & Quadrant III & Quadrant IV \\ (x, y) & (x, y) & (x, y) & (x, y) \\ (+, +) & (−, +) & (−, −) & (+, −) \end{matrix}$
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 $(0, b) .$
Solution of a Linear Equation
- An ordered pair $(x, y)$ is a solution of the linear equation $A x + B y = C$ , if the equation is a true statement when the x- and y- values of the ordered pair are substituted into the equation.

4.2 Graph Linear Equations in Two Variables

Graph a Linear Equation by Plotting Points
1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
2. Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
3. 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.
Graph a Linear Equation using the Intercepts
1. 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.
2. Step 2. Find a third solution to the equation.
3. Step 3. Plot the three points and then check that they line up.
4. Step 4. Draw the 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, 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 $a x + b y = c$ , 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=riserunm=riserun
1. Step 1. Locate two points on the line whose coordinates are integers.
2. Step 2. Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.
3. Step 3. Count the rise and the run on the legs of the triangle.
4. Step 4. Take the ratio of rise to run to find the slope.
Graph a Line Given a Point and the Slope
1. Step 1. Plot the given point.
2. Step 2. Use the slope formula $m = \frac{rise}{run}$ to identify the rise and the run.
3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
4. 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, $(0, b)$ is, $y = m x + b$ .
Graph a Line Using its Slope and y-Intercept
1. Step 1. Find the slope-intercept form of the equation of the line.
2. Step 2. Identify the slope and y-intercept.
3. Step 3. Plot the y-intercept.
4. Step 4. Use the slope formula $m = \frac{rise}{run}$ to identify the rise and the run.
5. Step 5. Starting at the y-intercept, count out the rise and run to mark the second point.
6. 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 $y = m x + b$ , graph by using the slope and y-intercept.
  Identify the slope and y-intercept and then graph.
- If the equation is of the form $A x + B y = C$ , find the intercepts.
  Find the x- and y-intercepts, a third point, and then graph.
Parallel lines are lines in the same plane that do not intersect.
- Parallel lines have the same slope and different y-intercepts.
- If m₁ and m₂ are the slopes of two parallel lines then $m_{1} = m_{2} .$
- Parallel vertical lines have different x-intercepts.
Perpendicular lines are lines in the same plane that form a right angle.
- If $m_{1} and m_{2}$ are the slopes of two perpendicular lines, then $m_{1} \cdot m_{2} = −1$ and $m_{1} = \frac{−1}{m_{2}}$ .
- 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
1. Step 1. Identify the slope.
2. Step 2. Identify the point.
3. Step 3. Substitute the values into the point-slope form, $y - y_{1} = m (x - x_{1})$ .
4. Step 4. Write the equation in slope-intercept form.
To Find an Equation of a Line Given Two Points
1. Step 1. Find the slope using the given points.
2. Step 2. Choose one point.
3. Step 3. Substitute the values into the point-slope form, $y - y_{1} = m (x - x_{1})$ .
4. 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 $y = m x + b$ .
- If given slope and a point, use point–slope form $y - y_{1} = m (x - x_{1})$ .
- If given two points, use point–slope form $y - y_{1} = m (x - x_{1})$ .
To Find an Equation of a Line Parallel to a Given Line
1. Step 1. Find the slope of the given line.
2. Step 2. Find the slope of the parallel line.
3. Step 3. Identify the point.
4. Step 4. Substitute the values into the point-slope form, $y - y_{1} = m (x - x_{1})$ .
5. Step 5. Write the equation in slope-intercept form.
To Find an Equation of a Line Perpendicular to a Given Line
1. Step 1. Find the slope of the given line.
2. Step 2. Find the slope of the perpendicular line.
3. Step 3. Identify the point.
4. Step 4. Substitute the values into the point-slope form, $y - y_{1} = m (x - x_{1})$ .
5. Step 5. Write the equation in slope-intercept form.

4.7 Graphs of Linear Inequalities

To Graph a Linear Inequality
1. Step 1. Identify and graph the boundary line.
  If the inequality is $\leq or \geq$ , the boundary line is solid.
  If the inequality is < or >, the boundary line is dashed.
2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
3. 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.