Equations – Section 4 Part B: Solving Multi-Step Equations Concept: Solving Multi-Step Equations Name: You should have completed Equations – Section 4 Part A: Solving Multi-Step Equations before beginning this handout. Warm Up Solve each multi-step equation below. Show all your steps and make sure you check to see if your solution is correct. 1. -t) +11) 4(t - 2) - (t + 3) = t - 1 4t - 8 - t 3t 3t - t 2t 2t - 11 ÷2) Check: L.S. = = = = = 2. -2.9x) -5.4) ÷2.1) Check: L.S. = = = = - 3 - 11 - 11 - 11 + 11 2t 2t 2 t 4(t - 2) - (t + 3) 4(5 - 2) - (5 + 3) 4(3) - (8) 12 - 8 4 5 x + 5.4 5 x - 2.9x + 5.4 2.1x + 5.4 2.1x + 5.4 - 5.4 2.1x 2.1x 2.1 x 5x + 5.4 5(-3) + 5.4 -15 + 5.4 -9.6 = = = = = = = t - 1 t - 1 t - t - 1 -1 -1 + 11 10 10 2 = 5 R.S. = t - 1 = 5 - 1 = 4 L.S. equals R.S., the solution is t = 5. = 2.9 x - 0.9 = = = = = = 2.9 x - 2.9x - 0.9 - 0.9 - 0.9 - 5.4 - 6.3 - 6.3 2.1 -3 R.S. = = = = 2.9 x - 0.9 2.9 (-3) - 0.9 -8.7 - 0.9 -9.6 L.S. equals R.S., the solution is x = -3. www.neufeldlearning.com 1 Equations – Section 4 Part B: Solving Multi-Step Equations COMPUTER COMPONENT Instructions: Login to UMath X Hover over the strand: Equations Select the section: Solving Multi-Step Equations NOTE: You will need to use the Menu feature of the program (found on the left side of your screen) in order to get to the lesson where you left off. Work through all Sub Lessons of the following Lessons in order: Summary Literal Equations As you work through the computer exercises, you will be prompted to make notes in your notebook/math journal. NOTES: Fill in the following: 1. Two ways to solve an equation. (a) Solve and equation with algebra tiles. (b) Solve an equation algebraically. 2. To keep a balance balanced, you must perform the same operation to both sides. 3. You know when you have a solution when: (a) There is one variable tile on one side of the equation (tile solution). (b) There is one variable on one side of the equation (algebraic solution). 4. Combine like terms if they are on the same side of the equation. 5. Equations with fractions, require you to first multiply each side (three words) by a common denominator (two words). This keeps the equation balanced. www.neufeldlearning.com 2 Equations – Section 4 Part B: Solving Multi-Step Equations 6. Use the original equation to check your answer by substituting your solution for the variable . Check each side of the equation. Your solution is correct if you have the same value on each side. Literal Equations 7. Are the following perimeter equations the same? Why or why not. 2 1 + and -2W) P P – 2W P – 2W = 2L + 2W = 2L + 2W - 2W = 2L ÷2) P – 2W 2 = P – 2W 2 = L 1 2L 21 SAME The perimeter equations are the same but in different forms. The first (1) is useful for finding P when the length and width are known. The second (2) is useful for finding L when the Perimeter and width are known. Solving Linear Equations: (You use similar steps to solve literal equations as you do for equations with one variable) Solve 4 x + 2 y = 16 for y The equation has two different variables, x and y Determine the variable you have to solve for: y. You will need to isolate the variable to solve the literal equation. www.neufeldlearning.com 3 Equations – Section 4 Part B: Solving Multi-Step Equations Fill in the blanks. Literal Equation Similar Equation 4 x + 2 y = 16 35 + 8y = 11 You will need to isolate the variable to solve the literal equation. 4x -4x + 2 y = 16 - 4x 35 - 35 + 8y = 11 - 35 Simplify 2 y = 16 - 4x 8y = - 24 Isolate y 2 y = 16 - 4x 2 2 8y = 8 -24 8 y = -3 = 16 - 4x 2 2 Simplify y = 8 - 2x www.neufeldlearning.com 4 Equations – Section 4 Part B: Solving Multi-Step Equations Literal Equations: Use the Frayer Diagram to demonstrate your understanding of the meaning of the word “Literal Equations”. First fill in examples and then the non- examples. Using these, determine the characteristics of “Literal Equations”. With the information in the chart, write your definition of “Literal Equations”. (Answers will vary) Example: Frayer Diagram Definition Characteristics - At least two variables equation An equation that is expressed by means of at least 2 different variables. Literal Equations Examples Non-Examples P = 2L + 2W V = BH 3 2x + 3 = 1 5y A = l ×w 4(t - 2) - (t + 3) = t - 1 d = 2r www.neufeldlearning.com 5 Equations – Section 4 Part B: Solving Multi-Step Equations OFF COMPUTER EXERCISES 1. Solve the following equations. (Remember to show your work and check your answers.) (a) 9 + 3(m - 4) = 5m + 1 Expand: Simplify: -3m) 9 + 3m - 12 3m - 3 3m - 3m - 3 -3 -3 - 1 -4 -4 2 -2 -1) ÷2) = = = = = = = = 5m 5m 5m 2m 2m 2m 2m 2 m + + + + 1 1 3m + 1 1 1 - 1 Check: L.S. = = = = = 9 + 3(m - 4) 9 + 3 ( -2 - 4 ) 9 + 3(-6) 9 - 18 -9 R.S. = = = = 5m + 1 5 (-2) + 1 -10 + 1 -9 L.S. equals R.S., the solution is m = -2. (b) 3 m - 4 ( m + 6 ) = 2 ( m + 2 ) - 13 Expand: -2m) +24) ÷-3) 3 m - 4m -1m -1 m - 2m -3m -3 m + 24 - 24 - 24 - 24 - 24 - 24 -3m -3m -3 m = = = = = = = = 2m + 4 - 13 2m - 9 2m - 2m - 9 -9 -9 + 24 15 15 -3 -5 Check: L.S. = = = = = 3m - 4(m + 6 ) R.S. = 2(m + 2) - 13 3m - 4(-5 + 6 ) = 2(-5 + 2) - 13 15 - 4 (1) = 2 (-3) - 13 -15 - 4 = -6 - 13 -19 = -19 L.S. equals R.S., the solution is m = -5. www.neufeldlearning.com 6 Equations – Section 4 Part B: Solving Multi-Step Equations (c) 3 - 2 ( x + 4 ) = -3 ( 1 - 2 x ) + 14 Expand: +2x) -11) ÷8) 3 - 2x - 8 - 2x - 5 - 2x + 2x - 5 -5 - 5 - 11 - 16 - 16 = = = = = = = 3 + 6x 6x 8x 8x 8x 8x 8 -2 = x 6 x + 14 + 11 + 2x + 11 + 11 + 11 - 11 8 Check: L.S. = = = = = 3 3 3 3 -1 (d) 2(x + 4) 2(-2 + 4) 2(2) 4 R.S. = -3(1 - 2 x) + 14 = -3(1 - 2 (-2)) + 14 = -3(1 + 4) + 14 = -3 ( 5 ) + 14 = -15 + 14 = -1 L.S. equals R.S., the solution is x = -2. 6 + 3(m - 4) = 6m - 3 Expand: -3m) +3) ÷3) 6 + 3m - 12 3m - 6 3m - 3m - 6 -6 -6 +3 -3 -3 = = = = = = = 6m - 3 6m - 3 6 m - 3m - 3 3m - 3 3m - 3 + 3 3m 3m 3 -1 = m 3 Check: L.S. = = = = = 6 + 3(m - 4) 6 + 3(-1 - 4) 6 + 3(-5) 6 - 15 -9 R.S. = = = = 6m - 3 6(-1) - 3 -6 - 3 -9 L.S. equals R.S., the solution is m = -1. www.neufeldlearning.com 7 Equations – Section 4 Part B: Solving Multi-Step Equations (e) 2n - 3 2 = -n - 1 4 2n - 3 21 = 1 2 (2n – 3) = -n - 1 4n – 6 = -n - 1 4n + n – 6 = -n + n - 1 5n – 6 = -1 5n – 6 + 6 = -1 + 6 5n = 5 5n 5 = = 5 5 n = 1 R.S. = -n - 1 4 = -(1) - 1 4 = -2 4 = -1 2 Clear Fraction: ×4) Expand: +n) +6) ÷5) 2 4 4 -n - 1 41 Check: L.S. = 2n - 3 2 = 2(1) - 3 2 = 2 - 3 2 = -1 2 L.S. equals R.S., the solution is n = 1. www.neufeldlearning.com 8 Equations – Section 4 Part B: Solving Multi-Step Equations (f) ×30) 30× 6 3 5 - x 3 = x 2 3 5 - x 3 = x × 30 2 = x × 30 2 10 30 × 3 5 1 +10x) 1 1 18 - 10x = 15x 18 - 10x + 10x = 15x 18 = 18 25 = 25x 25 18 25 = x ÷25) Check: LS = 15 - 30 × x 3 3 - 1 × 18 5 3 1 25 = 3 × 5 - 6 5 × 5 25 = 15 - 6 25 25 = 9 25 6 + 10x 25x RS = 18 × 1 25 2 = 9 25 L.S. equals R.S., the solution is x = 18. www.neufeldlearning.com 9 Equations – Section 4 Part B: Solving Multi-Step Equations g) 3 (2x - 1) 4 ×12) = 5 (2 – 4x) 6 3 2 12 × 3(2x - 1) = 4 12 × 5 (2 – 4x) 6 1 9(2x - 1) = 10 (2 - 4x) 18x - 9 = 20 - 40x Expand: +9) 1 18x - 9 + 9 = 18x = 18x + 40x = 58x = +40x) ÷58) 20 - 40x + 9 29 - 40x 29 - 40x + 40x 29 58x 58 = 29 58 x = 1 2 Check L.S. = 3 (2x - 1) 4 = 3 (2(1) - 1) 4 2 R.S. = 5 (2 – 4x) 6 = 5 (2 – 4(1) ) 6 2 1 2 1 1 = 3 (0) 4 = 5 (0 ) 6 = 0 = 0 L.S. equals R.S., the solution is x = 1. www.neufeldlearning.com 10 Equations – Section 4 Part B: Solving Multi-Step Equations h) ×12) 6a - 5 - 2 = 3 12 × 6a - 5 - 2 3 = 5a - 1 4 12 × + 1 3 5a - 1 4 4 3 12 × (6a - 5) - 12 × 2 = 3 12 × (5a - 1) 4 4 1 -15a) ÷9) Check: L.S. = = = = = = = = 6a - 5 - 2 3 6(5) - 5 - 2 3 30 - 5 - 2 3 25 - 2 3 25 - 2 × 3 3 1×3 25 - 6 3 19 3 19 3 + 12 × 1 3 1 24a - 20 - 24 24a - 44 24a - 44 + 44 24a 24a - 15a 9a 9a 9 a +44) + 1 3 = = = = = = = = 15a 15a 15a 15a 15a 45 45 9 5 R.S. 1 - 3 + 4 + 1 + 1 + 44 + 45 - 15a + 45 = = = = = = = = 5a - 1 + 4 5(5) - 1 + 4 25 - 1 + 4 24 + 1 4 3 6 + 1 1 3 6×3 + 1×3 1 3 1 3 1 3 1 3 18 + 1 3 19 3 L.S. equals R.S., the solution is a = 5. www.neufeldlearning.com 11 Equations – Section 4 Part B: Solving Multi-Step Equations 2. Solve each literal question. a) Solve: y = m x + b for x y y–b y–b m y–b m -b) ÷m) b) = = = = mx + b mx + b - b mx m x Solve: C = 2 p r + w C C - w C- w C- w 2r C- w 2r -w) ÷2r) c) = = = = = for p 2pr + w 2pr + w - w 2pr 2pr 2r p Solve: P = 2 L + 2 W for L (Hint: use a diagram.) W L -2W) L ÷2) W d) P P - 2W P - 2W P - 2W 2 P - 2W 2 = = = = = 2L + 2W 2L + 2W - 2W 2L 2L 2 L As you know, P = 2 L + 2 W is the formula for perimeter. If a field has a width of 25m and a perimeter of 206 m, find the length by using your answer in (c). (Hint: use a diagram.) 25m L P = 2L + 2W = 206 m www.neufeldlearning.com 12 Equations – Section 4 Part B: Solving Multi-Step Equations P - 2W 2 206 - 2(25) 2 206 - 50 2 156 2 78 = L = L = L = L = L Therefore the length of the field is 78m. www.neufeldlearning.com 13

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