Lesson #5 More on Solving Equations
In Lesson #4, we worked on solving equations which used the variable in only one location. Many common situations use more complex equations, however. Nonetheless, the same skills used so far will serve to solve any possible linear equation.
To solve linear equations:
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1) Remove parentheses by multiplication; 2) Collect like terms on each side; 3) Use addition or subtraction to remove the variable term from one side; 4) Use addition or subtraction to remove the constant term from the side of the remaining variable; 5) Divide both sides by the coefficient of the variable term. |
Keep in mind that each step is a process of simplification. Our goal is to simplify the equation to a new equation of the form "variable = number", which gives the solution to the original equation. Thus every step you take should result in a simpler equation (usually, fewer terms). Note that the process for solving equations is "working backwards" in the sense that we must undo the various operations that are applied to the variable. To evaluate a variable expression, we must follow the Order of Operations. To solve an equation, we have to remove those operations, one by one, in the reverse order.
Example 1. Solve 5x - 4 = 2x - 10.
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5x - 4 |
= 2x - 10 |
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Solution: |
-2x |
-2x |
(Step 3 above: subtract 2x from each side to remove the variable term from the right side) |
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3x - 4 |
= -10 |
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+4 |
+4 |
(Step 4: adding 4 to both sides removes the constant term from the variable side) |
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3x |
= -6 |
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(Step 5: divide both sides by 3 to remove the multiplier on the variable term) |
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x = -2 |
(Simplify the fractions to obtain the solution) |
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You can check the solution to any equation by substituting your answer into the original equation.
5x - 4 = 2x - 10
5(-2) - 4 = 2(-2) - 10
-10 - 4 = -4 - 10
-14 = -14
Since both sides are the same when x = -2, you know that you have correctly solved the equation.
Example 2. Solve 7y - 2(3y - 5) = 3(3y + 1) - 9.
Solution: 7y - 6y + 10 = 9y + 3 - 9 (1) Multiply through the parentheses.
(2) Collect like terms on each side.
At this point you should have at most
1y + 10 = 9y - 6 two terms on each side. Now continue as
in example 1.
-9y -9y (3) Remove the variable from one side.
-8y + 10 = -6
-10 -10 (4) Remove the constant from the other
-8y = -16 side.
(5) Divide off the variable coefficient.
This is the solution.
Exercises
Solve each of the following equations.1) 8x + 3 = 7x - 9 2) 3y + 2 = 2y + 13
3) 5x - 9 = 3x + 1 4) 4t - 28 = 7t - 1
5) 10a + 2 = 6a - 18 6) 12x - 7 = 15x - 4
7) 6y + 8 = y + 8 8) 5t - 2 = 4t + 26
9) 4Q - 2Q + 3 = 7Q + 13 10) 100B - 87 = 60B + 33
11) 3x - 2(6x + 3) = x - 4 12) 3(x - 4) = 5(x + 2)
13) 2(4y - 3) - 4 = 3(2y + 1) 14) 9 - 2(5f - 1) = -8
15) 6 + 2(9x + 5) = -4(x - 3) + 9 16) 3 - 3(3x + 2) = 4 - (2x + 5)
17) 8y - 4(2y - 7) + 4 = 3y - 7 18) 2x + 3(2 - 3x) = 14x - 6
19) 7w + 2(3w - 8) + 12 = 3(w - 1) + 5 20) 2(2x + 3) = 3(5x - 2) - 2(x + 8)
Advanced formula problems
The formula problems in Lesson #2 all asked you to find the quantity that is shown alone on one side of the formula. In more advanced problems, you may be asked to find one of the other quantities in the formula. Begin solving by writing the formula and identifying all the variables as before. After placing the known values in the formula, there should be one remaining unknown (variable) in the resulting equation. In this case you can simplify each side and then solve for the remaining variable, which should be the answer to the question.
Example 3. A rectangle has a length of 12 meters and a perimeter of 78 meters. Find the width of the rectangle.
Solution: Since the problem is about the perimeter of a rectangle, use the formula 
, where P = 78 is the perimeter and L = 12 is the length. So the equation becomes
78 = 2(12) + 2W
Here W is the remaining variable, so solve for W:
78 = 24 + 2W
78 - 24 = 2W
54 = 2W
W = 27.
The width of the rectangle is 27 meters.
More exercises
(Refer to Lesson #2 for the formulas if necessary)21. A coat that regularly sells for $125 is marked with a sale price of $75. What percent discount is this?
22. A triangle with a base of 21 inches has an area of 52.5 square inches. What is the height of the triangle?
23. A train traveled 318 kilometers in 3 hours. How fast was it going?
24. A savings account pays 7.5% simple interest. How much money would have to be deposited in order earn $87 interest in one year?
25. If you deposit $1500 in a savings account earning 5% simple interest, how long will it take to earn $37.50?
26. At a 35% off sale, a customer buys a lawnmower for $110. What was the regular price of the lawnmower?
27. A rock is thrown upward from a height of 50 feet. After 5 seconds, it hits the ground (h = 0). How fast was the rock initially thrown?
Check the answers to Lesson 5 exercises
Go on to Lesson 6
