Graphing Linear Equations Lesson 4-5 Graphing Linear Equations

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Objectives Determine whether an equation is linear Graph linear equations

Vocabulary Linear equation – an equation of a line (when graphed) Standard form – an equation of a line in the form Ax + By = C x-intercept – x-coordinate where it crosses the x-axis (when y = 0) y-intercept – y-coordinate where it crosses the y-axis (when x = 0)

Linear Equations Standard Form: Ax + By = C To find the y-intercept (let x = 0) By = C y = (C/B) To find the x-intercept (let y = 0) Ax = C x = (C/A) Y-intercept Form: y = mx + b (how calculator needs them) where m is called the slope and b is called the y-intercept (the slope is a rate of change: ∆y / ∆x)

Example 1a Determine whether 5x + 3y = z + 2 is a linear equation. If so, write the equation in standard form. First rewrite the equation so that the variables are on the same side of the equation. Original equation Subtract z from each side. Simplify. Since has 3 different variables, it cannot be written in the form Answer: This is not a linear equation.

Example 1b Determine whether is a linear equation. If so, write the equation in standard form. Rewrite the equation so that both variables are on the same side. Original equation Subtract y from each side. Simplify.

Example 1b cont To write the equation with integer coefficients, multiply each term by 4. Original equation Multiply each side of the equation by 4. Simplify. The equation is now in standard form where Answer: This is a linear equation.

Example 1c Determine whether 3x – 6y = 27 is a linear equation. If so, write the equation in standard form. Since the GCF of 3, 6, and 27 is not 1, the equation is not written in standard form. Divide each side by the GCF. Original equation Factor the GCF. Divide each side by 3. Simplify. Answer: The equation is now in standard form where

Example 1d Determine whether is a linear equation. If so, write the equation in standard form. To write the equation with integer coefficients, multiply each term by 4. Original equation Multiply each side of the equation by 4. Simplify. Answer: The equation can be written as Therefore, it is a linear equation in standard form where

Example 2 Graph In order to find values for y more easily, solve the equation for y. Original equation Add x to each side. Simplify. Multiply each side by 2. Simplify.

Example 2 cont Select five values for the domain and make a table. Then graph the ordered pairs. Answer: x 2 + 2x y (x,y) -3 2 + 2(-3) -4 (-3,-4) -1 2 + 2(-1) (-1,0) 2 + 2(0) 2 (0,2) 2 + 2(-2) 6 (2,6) 3 2 + 2(3) 8 (3,8) When you graph the ordered pairs, a pattern begins to form. The domain of y = 2 + 2x is the set of all real numbers, so there are an infinite number of solutions of the equation. Draw a line through the points. This line represents all the solutions of y = 2 + 2x.

Example 3 Shiangtai walks his dog 2.5 miles around the lake every day. Graph m = 2.5d, where m represents the number of miles walked and d represents the number of days walking. Select five values for d and make a table. Graph the ordered pairs and connect them to draw a line. d 2.5d t (d, t) 2.5(0) (0, 0) 4 2.5(4) 10 (4, 10) 8 2.5(8) 20 (8, 20) 12 2.5(12) 30 (12, 30) 16 2.5(16) 40 (16, 40)

Example 4 Determine the x-intercept and the y-intercept of 4x – y = 4. Then graph the equation. To find the x-intercept, let . Original equation Replace y with 0. Divide each side by 4. To find the y-intercept, let . Original equation Replace x with 0. Divide each side by –1.

Example 4 cont Answer: The x-intercept is 1, so the graph intersects the x-axis at (1, 0). The y-intercept is –4, so the graph intersects the y-axis at (0, –4). Plot these points. Then draw a line that connects them.

Summary & Homework Summary: Homework: Standard from: Ax + By = C, where A  0 and A and B are not both zero, and A, B, and C are integers whose greatest common factor is 1 To find the x-intercept, let y = 0. To find the y-intercept, let x = 0. Homework: pg 221; 16-36 even