1. PRE-ALGEBRA

2. Lesson 8-2 Warm-Up

3. PRE-ALGEBRA Equations with Two Variables (8-2) What is a “solution”? How do you find a solution when given one of the two variables of the equation? solution: an ordered pair that makes an equation with two variables a true statement (in other words, the graph of the equation will pass through that point) Example: (1, 2) is a solution for y = 2x, because both sides of the equation are equal (“true statement”) for x = 1, y = 2. [2 = 2(1) or 2 = 2  ] To find the solution to an equation when given one of its two variables, substitute the variable you know into the equation and solve for the other variable. Example: Solve y = 3x + 4 for x = -1. y = 3x + 4 y = 3(-1) + 4Substitute x = -1 into the equation. y = -3 + 4Simplify y = 1Solve for y. A solution for the equation y = 3x + 4 is (-1, 1).

4. PRE-ALGEBRA Find the solution of y = 4x – 3 for x = 2. y = 4x – 3 y = 4(2) – 3Replace x with 2. y = 8 – 3Multiply. y = 5Subtract. A solution of the equation is (2, 5). Equations With Two Variables LESSON 8-2 Additional Examples

5. PRE-ALGEBRA The equation a = 5 + 3p gives the price for admission to a park. In the equation, a is the admission price for one car with p people in it. Find the price of admission for a car with 4 people in it. a = 5 + 3p a = 5 + 3(4)Replace p with 4. a = 5 + 12Multiply. a = 17Add. A solution of the equation is (4, 17). The admission price for one car with 4 people in it is $17. Equations With Two Variables LESSON 8-2 Additional Examples

6. PRE-ALGEBRA Equations with Two Variables (8-2) What is a “linear equation”? How can you use a graph to fine the solutions to a linear equation? linear equation: an equation in which every solution (ordered pair that makes it a true statement) forms a line on a graph Example: any equation in which the x is to the first power (i.e. not x 2, x 3, x 4, etc.) will form a line when you graph its solutions To find the solutions to an equation using a graph: 1. create a function table by making up your own x values to find the y values of at least three points on the graph; 2. plot the points, and 3. draw a line through the points. Any ordered pairs on the line you drew are also solutions to the equation. Example: Graph y = - x + 3. Is (2,2) a solution? Step 1: Make a table of values to find at least three ordered pair solutions? 1212

7. PRE-ALGEBRA Equations with Two Variables (8-2) Step 2: Plot the ordered pair and draw a line through the points. Check: Substitute (2, 2) into the equation to make sure it makes a true statement. y = - x + 3 2 = - (2) + 3Substitute 2 for x and 2 for y. 2 = -1 + 3 2 = 2  True statement! The point (2, 2) is on the line, so it is a solution to the equation. 1212 1212

8. PRE-ALGEBRA x4x – 2(x, y) –24(–2) – 2 = – 8 – 2 = –10(–2, –10) 04(0) – 2 = 0 – 2 = –2(0, –2) 24(2) – 2 = 8 – 2 = 6(2, 6) Graph the ordered pairs. Draw a line through the points. Graph y = 4x – 2. Make a table of values to show ordered-pair solutions. Equations With Two Variables LESSON 8-2 Additional Examples

9. PRE-ALGEBRA Equations with Two Variables (8-2) What if the graph of an equation is a vertical or horizontal line? Example: Is y = 2 a function? Example: Is x = 2 a function? This is a horizontal line. In other words, for every value of x, y = 2. Since there is only one x value for every y value, the equation is a function. This is a vertical line. In other words, for every value of y, y = 2. Since there are an infinite number of points at x = 2 (doesn’t pass the vertical line test), the equation is NOT a function.

10. PRE-ALGEBRA For every value of x, y = –3. Graph each equation. Is the equation a function? This is a horizontal line. a.y = –3b.x = 4 The equation y = – 3 is a function. This is a vertical line. The equation y = 4 is not a function. For every value of y, x = 4. Equations With Two Variables LESSON 8-2 Additional Examples

11. PRE-ALGEBRA Equations with Two Variables (8-2) How do you graph an equation not in y = form? If the equation isn’t in y = form, you can solve for y before creating a function table.. Example: Graph 3x + y = -5. Step 1: Solve for y. 3x + y = -5Given 3x + y = -5Subtract 3x from both sides. - 3x -3x 0 + y = -3x -5 y = -3x -5 Simplify. Step 2: Make a table of values to find at least three ordered pair solutions. Step 3: Graph the points from your function table and draw a line through them.

12. PRE-ALGEBRA Solve y – x = 3 for y. Then graph the equation. 1212 Solve the equation for y. y – x = 3 1212 y = x + 3Simplify. 1212 y – x + x = 3 + xAdd x to each side. 1212 1212 1212 1212 Equations With Two Variables LESSON 8-2 Additional Examples

13. PRE-ALGEBRA (continued) Graph.Make a table of values. xx + 3(x, y) –2(–2) + 3 = –1 + 3 = 2(–2, 2) 0(0) + 3 = 0 + 3 = 3(0, 3) 2(2) + 3 = 1 + 3 = 4(2, 4) 1212 1212 1212 1212 Equations With Two Variables LESSON 8-2 Additional Examples

14. PRE-ALGEBRA Find the solution for each equation for x = 2. 1.y = –2x + 5 2.y = 7x3.y = 3x – 9 Solve each equation for y. Then graph each equation. 4.y – 2x = 3 5.2x + 2y = 8 (2, 1) (2, 14)(2, –3) y = 2x + 3y = –x + 4 Equations With Two Variables LESSON 8-2 Lesson Quiz