Pre-Algebra Point-Slope Form

Learn to find the equation of a line given one point and the slope.

point-slope form Vocabulary

Point on the line (x 1, y 1 ) Point-slope form y – y 1 = m (x – x 1 ) slope The point-slope of an equation of a line with slope m passing through (x 1, y 1 ) is y – y 1 = m(x – x 1 ).

Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. A. y – 7 = 3(x – 4) y – y 1 = m(x – x 1 ) y – 7 = 3(x – 4) m = 3 (x 1, y 1 ) = (4, 7) The line defined by y – 7 = 3(x – 4) has slope 3, and passes through the point (4, 7). The equation is in point-slope form. Read the value of m from the equation. Read the point from the equation. Example: Using Point-Slope Form to Identify Information About a Line

B. y – 1 = (x + 6) y – y 1 = m(x – x 1 ) (x 1, y 1 ) = (–6, 1) Rewrite using subtraction instead of addition y – 1 = (x + 6) y – 1 = [x – (–6)] 1 3 m = 1 3 The line defined by y – 1 = (x + 6) has slope, and passes through the point (–6, 1) Example: Using Point-Slope Form to Identify Information About a Line

Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. A. y – 5 = 2 (x – 2) y – y 1 = m(x – x 1 ) y – 5 = 2(x – 2) m = 2 (x 1, y 1 ) = (2, 5) The line defined by y – 5 = 2(x – 2) has slope 2, and passes through the point (2, 5). The equation is in point-slope form. Read the value of m from the equation. Read the point from the equation. Try This

B. y – 2 = (x + 3) 2 3 (x 1, y 1 ) = (–3, 2) Rewrite using subtraction instead of addition. 2 3 y – 2 = (x + 3) y – 2 = [x – (–3)] 2 3 m = 2 3 The line defined by y – 2 = (x + 3) has slope, and passes through the point (–3, 2) y – y 1 = m(x – x 1 ) Try This

Write the point-slope form of the equation with the given slope that passes through the indicated point. A. the line with slope 4 passing through (5, -2) y – y 1 = m(x – x 1 ) The equation of the line with slope 4 that passes through (5, –2) in point-slope form is y + 2 = 4(x – 5). Substitute 5 for x 1, –2 for y 1, and 4 for m. [y – (–2)] = 4(x – 5) y + 2 = 4(x – 5) Example: Writing the Point-Slope Form of an Equation

B. the line with slope –5 passing through (–3, 7) y – y 1 = m(x – x 1 ) The equation of the line with slope –5 that passes through (–3, 7) in point-slope form is y – 7 = –5(x + 3). Substitute –3 for x 1, 7 for y 1, and –5 for m. y – 7 = -5[x – (–3)] y – 7 = –5(x + 3) Example: Writing the Point-Slope Form of an Equation

Write the point-slope form of the equation with the given slope that passes through the indicated point. A. the line with slope 2 passing through (2, –2) y – y 1 = m(x – x 1 ) The equation of the line with slope 2 that passes through (2, –2) in point-slope form is y + 2 = 2(x – 2). Substitute 2 for x 1, –2 for y 1, and 2 for m. [y – (–2)] = 2(x – 2) y + 2 = 2(x – 2) Try This

B. the line with slope -4 passing through (-2, 5) y – y 1 = m(x – x 1 ) The equation of the line with slope –4 that passes through (–2, 5) in point-slope form is y – 5 = –4(x + 2). Substitute –2 for x 1, 5 for y 1, and –4 for m. y – 5 = –4[x – (–2)] y – 5 = –4(x + 2) Try This

A roller coaster starts by ascending 20 feet for every 30 feet it moves forward. The coaster starts at a point 18 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 150 feet forward. Assume that the roller coaster travels in a straight line for the first 150 feet. As x increases by 30, y increases by 20, so the slope of the line is or. The line passes through the point (0, 18) Example: Entertainment Application

y – y 1 = m(x – x 1 ) Substitute 0 for x 1, 18 for y 1, and for m. 2 3 The equation of the line the roller coaster travels along, in point-slope form, is y – 18 = x. Substitute 150 for x to find the value of y. 2 3 y – 18 = (150) 2 3 y – 18 = 100 y – 18 = (x – 0) 2 3 y = 118 The value of y is 118, so the roller coaster will be at a height of 118 feet after traveling 150 feet forward. Example Continued

A roller coaster starts by ascending 15 feet for every 45 feet it moves forward. The coaster starts at a point 15 feet above the ground. Write the equation of the line that the roller coaster travels along in point-slope form, and use it to determine the height of the coaster after traveling 300 feet forward. Assume that the roller coaster travels in a straight line for the first 300 feet. As x increases by 45, y increases by 15, so the slope of the line is or. The line passes through the point (0, 15) Try This

y – y 1 = m(x – x 1 ) Substitute 0 for x 1, 15 for y 1, and for m. 1 3 The equation of the line the roller coaster travels along, in point-slope form, is y – 15 = x. Substitute 300 for x to find the value of y. 1 3 y – 15 = (300) 1 3 y – 15 = 100 y – 15 = (x – 0) 1 3 y = 115 The value of y is 115, so the roller coaster will be at a height of 115 feet after traveling 300 feet forward. Try This Continued

Use the point-slope form of each equation to identify a point the line passes through and the slope of the line. 1. y + 6 = 2(x + 5) 2. y – 4 = – (x – 6) Write the point-slope form of the equation with the given slope that passes through the indicated point. 3. the line with slope 4 passing through (3, 5) 4. the line with slope –2 passing through (–2, 4) (–5, –6), 2 y – 5 = 4(x – 3) y – 4 = –2(x + 2) 2 5 (6, 4), – 2 5 Lesson Quiz