Math 8 Unit 7 Algebra and Graphing Relations Solving Equations Using Models We will be using algebra tiles to help us solve equations. We will practice showing work appropriately symbolically and pictorially using simple single step equations that you will easily be able to find the answer to without showing your work. However, you need to practice showing your work appropriately for future, more challenging problems! You will need to solve equations using addition, subtraction, multiplication and division, or a combination of these. To solve equations pictorially and concretely, we will be using algebra tiles. o Remember, yellow = positive and red = negative. (However, note that on the algebra tiles we have, yellow and green will be positive and red will be negative) Remember... a variable is a letter that represents an unknown value. A few things to remember... A constant is a value that is always the same. (This means that the value is constant / unchanging). o Ex. 2, -13, 6.7, 4 5, etc. A variable is a term whose value is unknown and not always the same. It is represented by a letter. o Ex. x, y, a, etc. A coefficient is a number that comes before a variable. It tells you how many of the variable you have. o Ex. 2x, 2 = coefficient, etc. EXAMPLE: Identify the variable, coefficient and constant terms in the following equation. 7x + 2 = 23 Coefficient = Variable = Constant = Variables are represented by this: Ones are represented by this:
Solving Equations with Addition and Subtraction 1. Model the equation using algebra tiles. 2. Make zero pairs on the side with the variable tile until all the positive and negative one tiles are cancelled out. 3. Remember to do the same thing to both sides of the equation! (Whatever you add to one side, be sure to add to the other!) 4. Determine the value of the variable. *To solve this question symbolically, record what you did using mathematical symbols. Verify your answer by checking. Example 1: x 3 = 4 Example 2: x + 4 = 9
Solving Equations with Division 1. Model the equation using algebra tiles. 2. Arrange the tiles into equal groups. (The number of groups is determined by the number of variable tiles). 3. The solution is the number of tiles in each group. *To solve this question symbolically, record what you did using mathematical symbols. Verify your answer by checking. Example 1: 2x = 4 Example 2: 3x = 15
Solving the Equation with Multiplication 1. Set up your equation pictorially. This means that you will have to divide your x tile into the number of portions specified by the denominator. Shade in one portion. Add your ones tiles, as you would normally. 2. Continue adding portions your portions of x (whether it s quarters, thirds, halves, etc.) until you have one whole x. Add the appropriate amount to the opposite side of the equation. Example #1 x 3 = 4 FOR EXAMPLE : if we use the form from above x = 3, you would 2 draw an x tile on the left side of the equation and shade half of it in. draw three ones tiles on the right side of the equation since you want to find one whole x, you would add another half x to the left side. since each half x is equal to three, add another three to the right side. your solution should end up being 6. Example #2 x 2 = 6
Solving one-step Equations Practice: Solve each equation pictorially, symbolically and then verify. x 2 = 3 x + 4 = 8 x 3 = 6
x + 4 = 6 3x = -9 4x = 12
2x = -6 5x = 10 x 2 = 2
x 4 = 2 x 3 = 3 x 2 = 4
Solving Multi-Step Equations We will use the example 2x + 3 = 11 to go through the steps. 1. Isolate the variable with its coefficient by adding or subtracting. 2. Determine the value of the variable by multiplying or dividing. Verify by Checking Again, substitute the value into the equation and solve. Remember to follow the proper order of operations. (BEDMAS) Rewrite the equation. Substitute in YOUR value for the variable. Solve.
Example 1: 2x + 4 = 10 Example 2: x 2 + 2 = 4
Math 8 Unit 7 Algebra and Graphing Equations Solving Algebraic Equations with Negative Coefficients Example 1: -x 2 = 4 Example 2: -2x = -4
Example 3: 2 = x 4 Example 4: -4x + 2 = 10
Example 5: - x 3 3 = -2 Math 8 Unit 7 Algebra and Graphing Equations The Distributive Property When you are given an equation like 2(x + 4) = 14, to solve symbolically: Example #1 2(x + 3) = 12 1. Multiply the number outside the brackets by both terms inside the brackets. 2. Isolate the variable (by adding or subtracting). 3. Solve for the variable (by multiplying or dividing). 4. Verify by substituting the answer back into the original equation.
Example #2 3(2x + 1) = 9 Your Assignment Complete the following questions, pictorially and symbolically on a separate sheet of paper. Verify all answers by checking. 1. 3x 4 = -7 2. 11 = 6x + 5 3. x + 5 = 9 4. n 4 = 8 5. x + 3 = -1 6. -3x = 6 Complete the following questions symbolically on a separate sheet of paper. Verify all answers by checking. 7. 4x = -36 8. 4x + 7 = 19 b 9. = 6 3 10. x 3 = -5 11. x 3 = 2 12. x 2 3 = -2 13. -3x + 2 = 14 14. 2(x 2) = 12 15. 2(x + 1) = 8 16. 3(x 2) = 3 17. 4( x + 1) = 4 18. 2(x + 2) = 2 19. 2( x 1) = 4 20. 3(x + 2) = 0 21. 2(2x + 1) = 10