5.3 Write Linear Equations in Point-Slope Form

Point-Slope Form y – y1 = m(x – x1) m = slope and it passes through (x1, y1) Y-intercept is not clear in this form unless you solve it for y.

Example Using Point-Slope Form write an equation for a line with a slope of ½ that goes through (7, -2).

Write an equation in point-slope form EXAMPLE 1 Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2. y – y 1 = m(x – x 1 ) Write point-slope form. y + 3 = 2(x – 4) Substitute 2 for m, 4 for x 1, and –3 for y 1.

Write an equation in point-slope form EXAMPLE 1 GUIDED PRACTICE for Example 1 Write an equation in point-slope form of the line that passes through the point (–1, 4) and has a slope of –2. 1. y – 4 = –2(x + 1) ANSWER

Graphing an Equation in Point-Slope Form y – y1 = m(x – x1) it goes through (x1, y1) To graph: Method 1: Change the signs on x1 and y1 then plot the point. Use the slope to find the next point.

Method 2 Solve for y and use slope-intercept form y – y1 = m(x – x1) Distribute m, add y1.

Graph an equation in point-slope form EXAMPLE 2 y + 2 = (x – 3). 2 3 Graph the equation SOLUTION Because the equation is in point-slope form, you know that the line has a slope of 2/3 and passes through the point (3, –2). Plot the point ( 3, –2 ). Find a second point on the line using the slope. Draw a line through both points.

Graph an equation in point-slope form EXAMPLE 2 y – 1 = (x – 2). – Graph the equation 2. GUIDED PRACTICE for Example 2 ANSWER

Example Graph y + 4 = 3(x – 1) Graph y – 1 = ½(x + 2)

Writing an Equation in Point-Slope Form Given Two Points Find the slope Plug either point into the formula There are two possible answers, but they are identical if solved for y. Example: Write an equation in point-slope form that goes through the points (-1, 3) and (1, 1)

Extra Example Write an equation in point slope form that goes through (-2, 3) and (1, -3)

Use point-slope form to write an equation EXAMPLE 3 GUIDED PRACTICE for Example 3 Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4). 3. y – 3 = (x – 2) or 1 2 y – 4 = (x – 4) 1 2 ANSWER

Solve a multi-step problem EXAMPLE 4 STICKERS You are designing a sticker to advertise your band. A company charges $225 for the first 1000 stickers and $80 for each additional 1000 stickers. Write an equation that gives the total cost (in dollars) of stickers as a function of the number (in thousands) of stickers ordered. Find the cost of 9000 stickers.

Solve a multi-step problem EXAMPLE 4 SOLUTION STEP 1 Identify the rate of change and a data pair. Let C be the cost (in dollars) and s be the number of stickers (in thousands). Rate of change, m: $80 per 1 thousand stickers Data pair (s 1, C 1 ): (1 thousand stickers, $225)

Solve a multi-step problem EXAMPLE 4 Write an equation using point-slope form. Rewrite the equation in slope-intercept form so that cost is a function of the number of stickers. STEP 2 C – C 1 = m(s – s 1 ) Write point-slope form. C – 225 = 80(s – 1) Substitute 80 for m, 1 for s 1, and 225 for C 1. C = 80s Solve for C.

Solve a multi-step problem EXAMPLE 4 Find the cost of 9000 stickers. C = 80(9) = 865 Substitute 9 for s. Simplify. ANSWER The cost of 9000 stickers is $865. STEP 3

Example A radio station charges $650 for the first minute of ad time and then $340 for each additional minute. Write an equation that gives the total cost (in dollars) to run an ad as a function of the number of minutes the ad runs. Find the cost of 7 minutes of ad time.

Is it Linear To tell if a situation can be modeled or graphed by a linear equation, the slope or rate of change must be constant.

Write a real-world linear model from a table EXAMPLE 5 WORKING RANCH The table shows the cost of visiting a working ranch for one day and night for different numbers of people. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of the number of people in the group Number of people Cost (dollars)

Write a real-world linear model from a table EXAMPLE 5 SOLUTION Find the rate of change for consecutive data pairs in the table. STEP – – 10 = – – 4 = 50, 450 – – 6 = 50, 550 – – 8 = 50, Because the cost increases at a constant rate of $50 per person, the situation can be modeled by a linear equation.

Write a real-world linear model from a table EXAMPLE 5 STEP 2 Use point-slope form to write the equation. Let C be the cost (in dollars) and p be the number of people. Use the data pair (4, 250). C – C 1 = m(p – p 1 ) Write point-slope form. C – 250 = 50(p – 4) Substitute 50 for m, 4 for p 1, and 250 for C 1. C = 50p +50 Solve for C.

Example The table shows the cost of renting a canoe for different times (in hours). Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of time (in hours). Time (hrs)Cost ($)

Guided Practice Top of page 305

Homework 1, 2 – 12 ev, 14 – 19, 20 – 34 ev, 38 – 42ev Bonus: 35, 36, 44