Objectives Find the two intercepts Graph a line using intercepts Rewrite equations in slope-intercept form. Identify the slope and y-intercept Graph a line using slope-intercept form. Solve systems of linear equations by graphing

The y-intercept is the y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0. The x-intercept is the x-coordinate of the point where the graph intersects the x-axis. The y-coordinate of this point is always 0.

Example 1A: Finding Intercepts Find the x- and y-intercepts. –3x + 5y = 30 To find the x-intercept, replace y with 0 and solve for x. To find the y-intercept, replace x with 0 and solve for y. –3x + 5y = 30 –3x + 5(0) = 30 –3x + 5y = 30 –3(0) + 5y = 30 –3x – 0 = 30 –3x = 30 0 + 5y = 30 5y = 30 x = –10 The x-intercept is (–10, 0). y = 6 The y-intercept is (0,6).

Example 1B Find the x- and y-intercepts. 4x + 2y = 16 To find the x-intercept, replace y with 0 and solve for x. To find the y-intercept, replace x with 0 and solve for y. 4x + 2y = 16 4x + 2(0) = 16 4x + 2y = 16 4(0) + 2y = 16 4x + 0 = 16 4x = 16 0 + 2y = 16 2y = 16 x = 4 The x-intercept is (4,0). y = 8 The y-intercept is (0,8)

Example 2: Sports Application Trish can run the 200 m dash in 25 s. The function f(x) = 200 – 8x gives the distance remaining to be run after x seconds. Graph this function and find the intercepts. What does each intercept represent? Neither time nor distance can be negative, so choose several nonnegative values for x. Use the function to generate ordered pairs. f(x) = 200 – 8x 200 25 5 10 20 x 160 120 40

Example 2 Continued Graph the ordered pairs. Connect the points with a line. y-intercept(0, 200). This is the number of meters Trish has to run at the start of the race. x-intercept: (25, 0). This is the time it takes Trish to finish the race {remaining distance is 0}.

Remember, to graph a linear function, you need to plot only two ordered pairs. It is often simplest to find the ordered pairs that contain the intercepts. Helpful Hint You can use a third point to check your line. Either choose a point from your graph and check it in the equation, or use the equation to generate a point and check that it is on your graph.

Use intercepts to graph the line –x + 3y = –6 Example 2 Use intercepts to graph the line –x + 3y = –6 x-intercept: –x = –6 y-intercept: 3y = –6 Plot (6, 0) and (0, –2). Connect with a straight line.

Example 3: Word Problem 3. An amateur filmmaker has $6000 to make a film that costs $75/h to produce. The function f(x) = 6000 – 75x gives the amount of money left to make the film after x hours of production. Graph this function and find the intercepts. What does each intercept represent? x-int.(80,0); number of hours it takes to spend all the money y-int.(0, 6000); the amount of money available.

Any linear equation can be written in slope-intercept form by solving for y and simplifying. In this form, you can immediately see the slope and y-intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

Example 4: Using Slope-Intercept Form to Graph Write the equation in slope-intercept form. Then graph the line described by the equation. y = 3x – 1 y = 3x – 1 is in the form y = mx + b slope: m = 3 = y-intercept: (0,–1) • Step 1 Plot (0, –1). • Step 2 Count 3 units up and 1 unit right and plot another point. Step 3 Draw the line connecting the two points.

Example 5: Using Slope-Intercept Form to Graph Write the equation in slope-intercept form. Then graph the line described by the equation. 2y + 3x = 6 Step 1 Write the equation in slope-intercept form by solving for y. 2y + 3x = 6 –3x –3x 2y = –3x + 6 Subtract 3x from both sides. Since y is multiplied by 2, divide both sides by 2.

Example 5 Continued Write the equation in slope-intercept form. Then graph the line described by the equation. Step 2 Graph the line. is in the form y = mx + b. • slope: m = y-intercept: (0, 3) • Plot (0, 3). • Count 3 units down and 2 units right and plot another point. • Draw the line connecting the two points.

Example 6 Write the equation in slope-intercept form. Then graph the line described by the equation. 6x + 2y = 10 Step 1 Write the equation in slope intercept form by solving for y. 6x + 2y = 10 –6x –6x 2y = –6x + 10 Subtract 6x from both sides. Since y is multiplied by 2, divide both sides by 2.

Example 6 Continued Write the equation in slope-intercept form. Then graph the line described by the equation. Step 2 Graph the line. • y = –3x + 5 is in the form y = mx + b. • slope: m = y-intercept: (0,5) • Plot (0, 5). • Count 3 units down and 1 unit right and plot another point. • Draw the line connecting the two points.

Example 4: Application A closet organizer charges a $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below.

b. Identify the slope and y-intercept and describe their meanings. Example 4 Continued A closet organizer charges $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below. b. Identify the slope and y-intercept and describe their meanings. The y-intercept is (0, 100). This is the cost for 0 hours, or the initial fee of $100. The slope is 30. This is the rate of change of the cost: $30 per hour. c. Find the cost if the organizer works 12 hrs. y = 30x + 100 Substitute 12 for x in the equation = 30(12) + 100 = 460 The cost of the organizer for 12 hours is $460.

A system of linear equations is a set of two or more linear equations containing two or more variables. A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. So, if an ordered pair is a solution, it will make both equations true.

All solutions of a linear equation are on its graph All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection. y = 2x – 1 y = –x + 5 The point (2, 3) is where the two lines intersect and is a solution of both equations, so (2, 3) is the solution of the systems.

  Example 5 Solve the system by graphing. Check your answer. y = –2x – 1 Graph the system. y = x + 5 The solution appears to be (–2, 3). Check Substitute (–2, 3) into the system. y = x + 5 y = –2x – 1 y = –2x – 1 3 –2(–2) – 1 3 4 – 1 3 3  y = x + 5 3 –2 + 5 3 3  (–2, 3) is the solution of the system.

Practice: Write each equation in slope-intercept form. Then graph the line described by the equation. 1. 6x + 2y = 10 2. x – y = 6 y = –3x + 5 y = x – 6

Practice: Systems Solve the system by graphing. 3. 4. Joy has 5 collectable stamps and will buy 2 more each month. Ronald has 25 collectable stamps and will sell 3 each month. After how many months will they have the same number of stamps? How many will that be? y + 2x = 9 (2, 5) y = 4x – 3 4 months 13 stamps