6.6 General Form of the Equation for a Linear Relation
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1 6.6 General Form of the Equation for a Linear Relation FOCUS Relate the graph of a line to its equation in general form. We can write an equation in different forms. y y 10 = 0 An equation for this graph is 5 y We can write this equation in slope-intercept form. 5 y 10 0 Add 10 to each side. 5 y 10 Subtract 5 from each side. y 5 10 Divide each side by. 5 y 5 When the equations are equivalent, they represent the same line. The equation 5 y 10 0 is in general form. All the terms are on the left side of the equation. The coefficients are integers, and the coefficient of is a positive integer. 5 y coefficients General Form of the Equation of a Line A By C 0 is the general form of the equation of a line, where A, B, and C are integers and A is positive Copyright 011 Pearson Canada Inc.
2 Eample 1 Rewriting an Equation in General Form Write each equation in general form. 1 a) y 5 b) y Solution a) y 5 Move all terms to the left side. y 5 0 Rearrange the terms so the -term is first. y 5 0 This equation is in general form. Remember to change the sign of a term when you move it to the other side of the equation. b) 1 y Remove the fraction. Multiply each term by. y a () b Simplify. y 6 y 6 0 y 6 0 Move all terms to the left side. Rearrange the terms so the -term is first. This equation is in general form. Check 1. Write this equation in general form: y 5 ( ) y 5 ( ) Remove the fraction. Multiply each side by. (y 5) ( ) ( ) ( ) Epand. y y 0 y 0 y 0 1( y ) 0 Move all terms to the left side. Collect like terms. Rearrange so the -term is first. Multiply each term by 1 so the -term is positive. This equation is in general form Copyright 011 Pearson Canada Inc. 59
3 Eample Finding the Slope of a Line with Its Equation in General Form Find the slope of the line with equation y 1 0. Solution Write the equation in slope-intercept form. y 1 0 Solve for y. Subtract from each side. y 1 Add 1 to each side. y 1 Divide each term by. y 1 Simplify. y Compare this equation with y m b. m So, the slope of the line is. Check 1. Find the slope of the line with equation 6 y 5 0. Write the equation in slope-intercept form. 6 y 5 0 y 5 y y Subtract from each side. Add to each side. Divide each term by. Simplify. y The slope of the line is Copyright 011 Pearson Canada Inc.
4 When you are given the equation of a line in general form, using intercepts may be the fastest way to graph the line. Eample Using Intercepts to Graph a Line Given in General Form a) Find the - and y-intercepts of the line y 8 0. b) Use the intercepts to graph the line. Solution a) y 8 0 To find the -intercept, substitute y 0. (0) 8 0 Solve for. 8 0 Add 8 to each side. 8 Divide each term by. 8 With practice, you can find the intercepts using mental math. The -intercept is. To find the y-intercept, substitute 0. (0) y 8 0 Solve for y. y 8 0 y 8 The y-intercept is 8. b) On a grid, mark a point at on the -ais and a point at 8 on the y-ais. Draw a line through the points. 8 y 6 + y 8 = Copyright 011 Pearson Canada Inc. 61
5 Check 1. a) Find the - and y-intercepts of the line y 0. y 0 To find the -intercept, substitute: _ ( ) 0 0 Solve for. Add to each side. Divide each term by. The -intercept is: To find the y-intercept, substitute: _ ( ) y 0 y 0 y Solve for. Add to each side. Divide each term by. y y The y-intercept is: b) Use the intercepts to graph the line. On a grid, mark a point at on the -ais and a point at on the -ais. Draw a line through the points y Copyright 011 Pearson Canada Inc.
6 Practice 1. Write each equation in general form. a) y 1 y 0 y 0 y 0 In general form, the equation is: 1 b) y () a 1 () b 0 0 Move all terms to the _ side. Put the -term first. Multiply by so the -term is positive. Multiply each term by. Simplify. Move all terms. Put the first. In general form, the equation is: c) y 1 ( ) 5 Multiply each side by. 5() () Epand. Move all terms. Collect _ terms. Put the first. In general form, the equation is:. Find the slope of each line. a) y 1 0 Write the equation in slope-intercept form. y 1 0 y 1 y Compare this equation with y m b. The slope of the line is: Subtract from each side. Add to each side Copyright 011 Pearson Canada Inc. 6
7 b) y 0 Write the equation in form. y 0 Subtract from each side. Subtract from each side. Divide each term by. The slope of the line is:. Find the - and y-intercepts of each line. a) 5y 0 0 To find the -intercept, substitute: _ 5( ) The -intercept is: To find the y-intercept, substitute: _ ( ) 5y 0 0 5y 0 5y 5y y The y-intercept is: b) 6y 18 0 To find the -intercept, substitute: _ To find the y-intercept, substitute: _ The -intercept is: The y-intercept is: Copyright 011 Pearson Canada Inc.
8 . Use intercepts to graph y 1 0. To find the -intercept, substitute: _ To find the y-intercept, substitute: _ The -intercept is: The y-intercept is: To graph the line: On the grid, mark a point at on the -ais and 6 y a point at on the -ais. Draw a line through the points Write this equation in slope-intercept form, then graph it: y 16 0 y 16 0 Solve for y. In slope-intercept form, the equation is: The slope is _. y The y-intercept is. 6 To graph the line: Plot a point at on the -ais, then move units and units. Mark a point. Draw a line through the points Copyright 011 Pearson Canada Inc. 65
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