Chapter 1 Linear Equations and Linear Functions

1.5 Finding Linear Equations

Example: Using Slope and a Point to Find an Equation of a Line Find an equation of the line that has slope m = 3 and contains the point (2, 5).

Solution The equation for a nonvertical line can be put into the form y = mx + b. Since m = 3, we have y = 3x + b

Solution To find b, recall that every point on the graph of an equation represents a solution of that equation. In particular, the ordered pair (2, 5) should satisfy the equation. Substitute –1 for b in y = 3x + b: y = 3x – 1

Solution Use TRACE on a graphing calculator to verify that the graph of y = 3x – 1 contains the point (2, 5).

Example: Using Two Points to Find an Equation of a Line Find an equation of the line that contains the points (–2, 6) and (3, –4).

Solution Find the slope of the line: Thus, we have y = –2x + b.

Solution Since the line contains the point (3, –4), substitute 3 for x and –4 for y: So, the equation is y = –2x + 2.

Solution We can use a graphing calculator to check that the graph of y = –2x + 2 contains both (–2, 6) and (3, –4).

Finding a Linear Equation That Contains Two Given Points To find an equation of the line that passes through two given points whose x-coordinates are different, 1. Use the slope formula, to find the slope of the line.

Finding a Linear Equation That Contains Two Given Points 2. Substitute the m value you found in step 1 into the equation y = mx + b. 3. Substitute the coordinates of one of the given points into the equation you found in step 2, and solve for b.

Finding a Linear Equation That Contains Two Given Points 4. Substitute the m value you found in step 1 and the b value you found in step 3 into the equation y = mx + b. 5. Use a graphing calculator to check that the graph of your equation contains the two given points.

Example: Finding the Equation of a Line Parallel to a Given Line Find an equation of a line l that contains the point (5, 3) and is parallel to the line y = 2x – 3.

Solution For the line y = 2x – 3, the slope is 2. So, the slope of the parallel line l is also 2. An equation of the line l is y = 2x + b. To find b, substitute the coordinates of (5, 3) into the equation y = 2x + b: An equation of l is y = 2x – 7.

Solution We use a graphing calculator to verify our equation.

Example: Finding the Equation of a Line Perpendicular to a Given Line Find an equation of a line l that contains the point (6, –7) and is perpendicular to the line –2x + 5y = 10.

Solution Isolate y in the equation –2x + 5y = 10:

Solution For the line the slope is The slope of the line l must be the opposite of the reciprocal of or An equation of l is

Solution To find b, substitute the coordinates of the given point (6, –7) into the equation: An equation of l is

Solution Use a graphing calculator to verify our work.

Point-Slope Form If a nonvertical line has slope m and contains the point (x1, y1), then an equation of the line is y – y1 = m(x – x1)

Example: Using Point-Slope Form to Find an Equation of a Line A line has slope m = 2 and contains the point (3, –8). Find an equation of the line.

Solution Substituting x1 = 3, y1 = –8, and m = 2 in the equation y – y1 = m(x – x1) gives

Example: Using Point-Slope Form to Find an Equation of a Line Use the point-slope form to find an equation of the line that contains the points (–5, 2) and (3, –1). Then write the equation in slope-intercept form.

Solution Begin by finding the slope of the line:

Solution Then we substitute x1 = 3, y1 = –1, and in the equation: