Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Quadratic equation wikipedia , lookup
Quartic function wikipedia , lookup
Cubic function wikipedia , lookup
System of polynomial equations wikipedia , lookup
Elementary algebra wikipedia , lookup
History of algebra wikipedia , lookup
Signal-flow graph wikipedia , lookup
PRE-ALGEBRA Lesson 8-2 Warm-Up PRE-ALGEBRA Equations with Two Variables (8-2) What is a “solution”? 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 ] How do you find a solution when given one of the two variables of the equation? 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) + 4 y = -3 + 4 y=1 Substitute x = -1 into the equation. Simplify Solve for y. A solution for the equation y = 3x + 4 is (-1, 1). PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Additional Examples Find the solution of y = 4x – 3 for x = 2. y = 4x – 3 y = 4(2) – 3 y=8–3 y=5 Replace x with 2. Multiply. Subtract. A solution of the equation is (2, 5). PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Additional Examples 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) a = 5 + 12 a = 17 Replace p with 4. Multiply. Add. A solution of the equation is (4, 17). The admission price for one car with 4 people in it is $17. PRE-ALGEBRA Equations with Two Variables (8-2) What is 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, x3, x4, etc.) will form a line when you graph its solutions How can you use a graph to fine the solutions to a linear equation? 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 = - 1 x + 3. Is (2,2) a solution? 2 Step 1: Make a table of values to find at least three ordered pair solutions? PRE-ALGEBRA Equations with Two Variables (8-2) Step 2: Plot the ordered pair and draw a line through the points. The point (2, 2) is on the line, so it is a solution to the equation. Check: Substitute (2, 2) into the equation to make sure it makes a true statement. y=-1 x+3 2 2 = - 12 (2) + 3 Substitute 2 for x and 2 for y. 2 = -1 + 3 2=2 True statement! PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Additional Examples Graph y = 4x – 2. Make a table of values to show ordered-pair solutions. x –2 0 2 4x – 2 4(–2) – 2 = – 8 – 2 = –10 4(0) – 2 = 0 – 2 = –2 (x, y) (–2, –10) (0, –2) 4(2) – 2 = 8 – 2 = 6 (2, 6) Graph the ordered pairs. Draw a line through the points. 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? 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. Example: Is x = 2 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. PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Additional Examples Graph each equation. Is the equation a function? a. y = –3 b. x = 4 For every value of x, y = –3. For every value of y, x = 4. This is a horizontal line. The equation y = – 3 is a function. This is a vertical line. The equation y = 4 is not a function. 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 = -5 3x + y = -5 - 3x -3x 0 + y = -3x -5 y = -3x -5 Step 2: Make a table of values to find at least three ordered pair solutions. Given Subtract 3x from both sides. Simplify. Step 3: Graph the points from your function table and draw a line through them. PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Additional Examples Solve y – 1 x = 3 for y. Then graph the equation. 2 Solve the equation for y. y– 1 1 x=3 2 1 1 y – 2 x + 2 x = 3 +2 1 y=2 x+3 x 1 Add 2 x to each side. Simplify. PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Additional Examples (continued) Make a table of values. 1 2 1 –2 2 1 0 2 1 2 2 x Graph. x+3 (x, y) (–2) + 3 = –1 + 3 = 2 (–2, 2) (0) + 3 = 0 + 3 = 3 (0, 3) (2) + 3 = 1 + 3 = 4 (2, 4) PRE-ALGEBRA Equations With Two Variables LESSON 8-2 Lesson Quiz Find the solution for each equation for x = 2. 1. y = –2x + 5 2. y = 7x (2, 1) (2, 14) 3. y = 3x – 9 (2, –3) Solve each equation for y. Then graph each equation. 4. y – 2x = 3 5. 2x + 2y = 8 y = 2x + 3 y = –x + 4 PRE-ALGEBRA