Solving Algebraic Equations An equation is a mathematical sentence that uses a equal sign (=).

An equation is an open sentence if it contains one or more variables.

Open sentences may be true or false depending on the values of its variable. Equations Tell whether each equation is true, false, or open. Explain. 1) -14n - 7 = 7

2) 7k - 8k = 15

3) 10 + (15) - 5 = 5

4) 32 ÷ (-4) + 6 = -72 ÷ 8 + 7 A solution of an equation containing a variable is a value of the variable that makes the equation true. Tell whether the given number is a solution of each equation. 5) -6 = 14 - 11n; 2 6) 7c - (-5) = 26; 3

7) 25 - 10z = 15; -1 8) -8a - 12 = -4; 1 Use a table to find the solution of each equation. 9) 5h - 13 = 12 10)8n + 16 = 24 Write an equation for each sentence. 12) A computer programmer works 40 hours per week. What is an equation that relates the number of weeks w that the programmer works and the number of hours h that the programmer spends working?

13) Josie is 11 years older than Macy. What is an equation that relates the age of Josie J and the age of Macy M? Use mental math to find the solution of each equation. 14) t - 7 = 10 15) 12 = 5 - h

16) 17) Use a table to find two consecutive integers between which the solution lies. 11) 7t - 20 = 33 One-Step Equations Two-Step Equations Multi-Step Equations Equations with Variables on Both Sides A Literal Equation is an equation that involves two or more variables. Literal Equations Equivalent equations are equations that have the same solutions.

To solve an equation, you must isolate the variable.

An inverse operation undoes another operation. Solve each equation using addition or subtraction. Check your answer. 1) 8 = a - 2 2) x + 7 = 11

3) r - 2 = -6 4) -18 = m + 12 Solve each equation using multiplication or division. Check your answer. 5) -3p = -48 6) -98 = 7t

7) 8) Solve each equation. Check your answer. 9) 10)

11) 12) Define a variable and write an equation for each situation. Then solve. 13) Susan’s cell phone plan allows her to use 950 minutes per month with no additional charge. She has 188 minutes left for this month. How many minutes has she already used this month?

14) In the fifth year of operation, the profit of a company was 3 times the profit it earned in the first year of operation. If its profit was $114,000 in the fifth year of operation, what was the profit in the first year? Define a variable and write an equation for each situation. Then solve. To solve a two-step equation, identify the operations and undo them using reverse operations. You can undo the operations in the reverse order of the order of operations. 1) Chip earns a base salary of $500 per month as a salesman. In addition to the salary, he earns $90 per product that he sells. If his goal is to earn $5000 per month, how many products does he need to sell? Solve each equation. Check your answer. Solve each equation. Check your answer. Solve each equation. Check your answer. 11) The fare for a taxicab is $5 per trip plus $0.50 per mile. The fare for the trip from the airport to the convention center was $11.50. Write and solve an equation to find how many miles the trip is from the airport to the convention center.

12) An online movie club offers a membership for $5 per month. Members can rent movies for $1.50 per rental. A member was billed $15.50 one month. Write and solve an equation to find how many movies the member rented. Describe, using words, how to solve the equation 6 - 4x = 18. List any properties utilized in the solution. Write an equation to model each situation. Then solve the equation. 1) General admission tickets to the fair cost $3.50 per person. Ride passes cost an additional $5.50 per person. Parking costs $6 for the family. The total costs for ride passes and parking was $51. How many people in the family attended the fair? Solve each equation. Check your answer. 2) 6(3m + 5) = 66 3) -5(x - 3) = -25 Solve each equation. Check your answer. Solve each equation. Check your answer. Solve each equation. Describe two different ways to solve: 4) Solve each equation. 1) 3n + 2 = –-2n -– 8 Solve each equation. 3) 7(h + 3) = 6(h - 3) Solve each equation. 5) -8x - (3x + 6) = 4 - x Write and solve an equation for each situation. Check your solution. 8) Shirley is going to have the exterior of her home painted. Tim’s Painting charges $250 plus $14 per hour. Colorful Paints charges $22 per hour. How many hours would the job need to take for Tim’s Painting to be the better deal? An equation that is true for every possible value of the variable is an identity.

For example x + 1 = x + 1 is an identity.

An equation has no solution if there is no value of the variable that makes the equation true.

For example x + 1 = x + 2 has no solution. Determine whether each equation is an identity or whether it has no solution. 1) 5x + 2x - 3 = -3x + 10x Solve each equation.

If the equation is an identity, write identity.

If it has no solution, write no solution. 2) -(3z + 4) = 6z - 3(3z + 2) Solve each equation.

If the equation is an identity, write identity.

If it has no solution, write no solution. 3) -2(j - 3) = -2j + 6 A square and a rectangle have the same perimeters. The length of a side of the square is 4x - 1. The length of the rectangle is 2x + 1 and the width is x + 2. Write and solve an equation to find x. Describe the difference between an equation that is defined as an identity and an equation that has no solution. Provide an example of each and explain why each example is an identity or has no solution. Solve each equation for m.

Then find the value of m for each value of n. 1) -5n = 4m + 8; n = -1, 0, 1

2) 2m = -6n - 5; n = 1, 2, 3 Solve each equation for m.

Then find the value of m for each value of n.

3) 8n = -3m + 1; n = -2, 2, 4

4) 4n - 6m = -2; n = -2, 0, 2 Solve each equation for x. 5) mx - 2nx = p

6) 6x + x = 3r Solve each equation for x. –7) -2(x + a) = 4x Solve each equation for x. Solve each equation for x. Describe and correct the error made in solving the literal equation below for n. Solve the problem.

Round to the nearest tenth, if necessary. Use 3.14 for pi. What is the radius of a circle with circumference 3 ft? Solve the problem.

Round to the nearest tenth, if necessary. Solve the problem.

Round to the nearest tenth, if necessary. What is the width of a rectangle with length 16 cm and area 128 cm? A rectangle has perimeter 200 in. and length 60 in. What is the width? Solve the problem.

Round to the nearest tenth, if necessary. Solve the problem.

Round to the nearest tenth, if necessary. Solve each equation for the given variable. Joan drives 333.5 miles before she has to buy gas. Her car gets 29 miles per gallon. How many gallons of gas did the car start out with? A triangle has base 5 m and area 25 m. What is the height? A) 4k + 2n = n -– 3; n

B) 3ab –- 2bc = 12; c Define a variable and write an equation for each situation. Then solve. 2) A pizza shop charges $9 for a large cheese pizza. Additional toppings cost $1.25 per topping. Heather paid $15.25 for her large pizza. How many toppings did she order? Solve each equation. 5) 6) 7) Solve each equation. 2) -–12 + 5k = 15 -– 4k 4) -(5a + 6) = 2(3a + 8) Solve each equation. 6) 14 + 3n = 8n - 3(n - 4) Solve each equation.

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# Solving Equations

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