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Introduction Often, an equation will be shown in standard form, such as 5x – 2y = 10. However, for a function to be graphed, its equation may need to be rearranged so it is in a more usable format. Typically, this means solving the equation for y. 1 2.1 Skill 1: Solving Equations in Standard Form for y Key Concepts • To solve an equation for y means to rearrange the equation so the y is by itself on the left side of the equal sign. For example, in the equation y = 2x + 5, the y is by itself and the rest of the equation shows what y equals. • The standard form of a linear function is written as ax + by = c, where a, b, and c are integers and a is positive. In this form, a and b are coefficients and c is the constant term. • The most common and efficient way to graph this type of equation is to put it in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. 2.1 Skill 1: Solving Equations in Standard Form for y 2 Key Concepts, continued • Notice that in order to put the equation in slopeintercept form, the equation must be solved for y. Solving an Equation for y 1. Move the ax term to the other side of the equation by adding its opposite to both sides of the equation. 2. Divide every term on both sides of the equation by b. Note: It is best to arrange the final answer so the xterm comes before the constant term. That way, the equation will be in slope-intercept form, y = mx + b. 3 2.1 Skill 1: Solving Equations in Standard Form for y Key Concepts, continued • If the value of y is known, the value of x can be found by substituting the y-value, and then solving the equation for x. Finding the Value of x Given the y-value 1. Move any terms that are added to or subtracted from the x-term by performing the opposite operation on both sides of the equation. (For example, if 6 is being added to x, subtract 6 from both sides.) 2. Divide both sides by the coefficient of x. 4 2.1 Skill 1: Solving Equations in Standard Form for y Guided Practice Example 1 Solve the equation 8x – 2y = 6 for y. 5 2.1 Skill 1: Solving Equations in Standard Form for y Guided Practice: Example 1, continued 1. Move the term with x to the other side of the equation by adding its opposite to both sides of the equation. The term with x is 8x. Move this term by adding the opposite of 8x to both sides of the equation. The opposite of 8x is –8x; therefore, add –8x to both sides of the equation. 6 2.1 Skill 1: Solving Equations in Standard Form for y Guided Practice: Example 1, continued 8x – 2y = 6 Original equation 8x – 2y + (–8x) = 6 + (–8x) Add –8x to both sides of the equation. –2y = 6 – 8x Simplify. The result of adding the opposite of 8x to both sides of the equation is –2y = 6 – 8x. 7 2.1 Skill 1: Solving Equations in Standard Form for y Guided Practice: Example 1, continued 2. Divide every term on both sides of the equation by the coefficient of y. The coefficient of y is –2. Therefore, divide every term on both sides of the equation by –2. –2y = 6 – 8x -2y -2 = 6 -2 - 8x y = –3 + 4x -2 Equation from the previous step Divide every term by –2. Simplify. 8 2.1 Skill 1: Solving Equations in Standard Form for y Guided Practice: Example 1, continued y = 4x – 3 Rearrange the terms on the right side so the x-term is listed first. Note that it is best to rearrange the terms so the x-term comes before the constant term. That way, the equation is in slope-intercept form. The equation 8x – 2y = 6 solved for y is y = 4x – 3. 9 2.1 Skill 1: Solving Equations in Standard Form for y Guided Practice: Example 1, continued 10 2.1 Skill 1: Solving Equations in Standard Form for y