Solving Basic Algebra Equations Practice Questions

Introduction

Solving equations is not very different from working with numerical or algebraic expressions. An equation is a mathematical statement where two expressions are set equal to each other. Using logic and mathematical operations, you can manipulate the equation to find a solution. Simply put, that is what you have done when you have the variable on one side of the equal sign and a number on the other. The answer explanations will show and identify all the steps you will need to solve basic equations. There will be different solutions to similar problems to show a variety of methods for solving equations. But they all rely on the same rules. Look over the Tips for Solving Basic Equations before you begin this lesson's questions.

Tips for Solving Basic Equations

If a number is being added to or subtracted from a term on one side of an equation, you can eliminate that number by performing the inverse operation.
The inverse of addition is subtraction. The inverse of subtraction is addition. If you add or subtract an amount from one side of the equation, you must do the same to the other side to maintain the equality.
The inverse of multiplication is division. If a variable is being multiplied by a coefficient, you can eliminate the coefficient by dividing both sides of the equation by that coefficient, leaving you with just one of the variables. If a is a number ≠ 0, then ax ÷ a = x.
The inverse of division is multiplication. If a number is dividing a variable, you can multiply the term by the number, leaving you with one of the variables. If b is a number ≠ 0, then .
You can perform any operation to a term in an equation. Just remember that you must do it to both sides. When you have the variable isolated on one side of the equation, the value on the other side is your solution.

Practice Questions

a + 21 = 32
x – 5 = 32
y + 17 = ^–12
b – 15 = 71
12 – c = ^–9
s– ^–4 = ^–1

m + 2(5 – 24) = ^–76
2a = 24
4x = ^–20
^–3y = 18
27b = 9
45r = ^–30
0.2c = 5.8
Jack paid $21,000 for his new car. This was the suggested selling price of the car. What was the suggested selling price of the car?
After putting 324 teddy bears into packing crates, there were 54 crates filled with bears. If each crate contained the same number of bears, how many bears were in each packing crate?
Only 3% of turtle hatchlings will live to become breeding adults. How many turtles must have been born if the current number of breeding adults is 1,200?
This year, a farmer planted 300 acres of corn. This was 1.5 times as many acres as he planted last year. How many acres did he plant this year?
A business executive received a $6,000 bonus check from her company at the end of the year. This was 5% of her annual salary. How much was her annual salary before receiving the bonus?

Answers

Numerical expressions in parentheses like this [ ] are operations performed on only part of the original expression. The operations performed within these symbols are intended to show how to evaluate the various terms that make up the entire expression.

Expressions with parentheses that look like this ( ) contain either numerical substitutions or expressions that are part of a numerical expression. Once a single number appears within these parentheses, the parentheses are no longer needed and need not be used the next time the entire expression is written.

When two pair of parentheses appear side by side like this ( )( ), it means that the expressions within are to be multiplied.

Sometimes parentheses appear within other parentheses in numerical or algebraic expressions. Regardless of what symbol is used, ( ), { }, or [ ], perform operations in the innermost parentheses first and work outward.

Underlined equations show the simplified result.

1. Subtract 21 from both sides of the equation.
Associate like terms.	a + (21 – 21) = (32 – 21)
Perform the numerical operation in the parentheses.	a + (0) = (11)
Zero is the identity element for addition.	a = 11
2. Add 25 to each side of the equation.	25 + x – 25 = 32 + 25
Use the commutative property for addition.	x + 25 – 25 = 32 + 25
Associate like terms.	x + (25 – 25) = (32 + 2/td>
Perform the numerical operation in the parentheses.	x + (0) = (57)
Zero is the identity element for addition.	x = 57
3. Subtract 17 from both sides of the equation.	y + 17 – 17 = –12 – 17
Associate like terms.	y + (17 – 17) = (–12 – 17)
Change subtraction to addition and change the sign of the term that follows.	y + (17 + –17) = (–12 + –17)
Apply the rules for operating with signed numbers.	y + (0) = –29
Zero is the identity element for addition.
3. Change subtraction to addition and change the sign of the term that follows.	b + –15 = 71
Add +15 to each side of the equation.	b + –15 + +15 = 71 + +15
Associate like terms.	b + (–15 + 15) = (71 + 15)
Apply the rules for operating with signed numbers.	b + (0) = 86
Zero is the identity element for addition.	b = 86
4. Change subtraction to addition and change the sign of the term that follows.	b + –15 = 71
Add +15 to each side of the equation.	b + –15 + +15 = 71 + +15
Associate like terms.	b + (–15 + 15) = (71 + 15)
Apply the rules for operating with signed numbers.	b + (0) = 86
Zero is the identity element for addition.	b = 86
5. Change subtraction to addition and change the sign of the term that follows.	12 + ^–c = ^–9
Add +c to each side of the equation.	12 + –c + ⁺c = ^–9 + ⁺c
12 + –c + ⁺c = ^–9 + ⁺c	12 + (–c + ⁺c) = ^–9 + ⁺c
	12 + (0) = –9 + c
	12 = ^–9 + c
Add +9 to each side of the equation.	⁺9 + 12 = ⁺9 + –9 + c
Associate like terms./td>	(⁺9 + 12) = (⁺9 + ^–9) + c
Apply the rules for operating with signed numbers.	(21) = (0) + c
Zero is the identity element for addition.	21 = c
6. Change subtraction to addition and change the sign of the term that follows.	s + ⁺4 = –1
Add –4 to each side of the equation.	s + ⁺4 + ^–4 = ^–1 + ^–4
Associate like terms.	s + (⁺4 + ^–4) = ^–1 + ^–4
Apply the rules for operating with signed numbers.	s + (0) = (^–5)
Subtracting zero is the same as adding zero.	s = –5
7. Add to each side of the equation.
Associate like terms.
Apply the rules for operating with signed numbers.
Zero is the identity element for addition.
8. Change subtraction to addition and change the sign of the term that follows.
Subtract from both sides of the equation.
Associate like terms.
Change subtraction to addition and change the sign of the term that follows.
Apply the rules for operating with signed numbers.
Change the improper fraction to a mixed number.
9. Change subtraction to addition and change the sign of the term that follows.
Use the distributive property of multiplication.
Perform the operation in parentheses.	c + (–2 + +20) = 20
Apply the rules for operating with signed numbers.	c + (–2 + +20) = 20
Subtract 18 from both sides of the equation.	c + (18) = 20
Associate like terms.	c + 18 – 18 = 20 – 18
Perform the operation in parentheses.	c + (18 – 18) = 20 – 18
Zero is the identity element for addition.	c = 2
10. Use the distributive property of multiplication.	m + (2 · 5 – 2 · 24) = ^–76
The order of operations is to multiply first.	m + (10 – 48) = ^–76
Add +38 to each side of the equation.	m + ^–38 + ⁺38 = ^–76 + ⁺38
Associate like terms.	m + (^–38 + ⁺³⁸) = ^–76 + ⁺38
Apply the rules for operating with signed numbers.	m + (0) = ^–38

11. Divide both sides of the equation by 2.	2a ÷ 2 = 24 ÷ 2
	a = 24 ÷ 2
Apply the rules for operating with signed numbers.	a = 12
Another method is as follows:
	a = 12
12. Divide both sides of the equation by 4.	4x ÷ 4 = –20 ÷ 4
	x = –20 ÷ 4
Apply the rules for operating with signed numbers.
Another look for this solution method is as follows:

13. Divide both sides of the equation by ^–3.

Apply the rules for operating with signed numbers.
14. Divide both sides of the equation by 27.

Reduce fractions to their simplest form.
15. Divide both sides of the equation by 45.

Reduce fractions to their simplest form (common factor of 15).
16. Divide both sides of the equation by 0.2.

Divide.	c = 29
17. Multiply both sides of the equation by 7.
	x = 7(16)
Multiply.	x = 112
18. Multiply both sides of the equation by –4.
	y = ^–4 · ^–12
Signs the same? Multiply and give the result a positive sign.	y = ⁺48 = 48
19. Divide both sides of the equation by

Dividing by a fraction is the same as multiplying by its reciprocal.
	a = 81
20. Divide both sides of the equation by

Dividing by a fraction is the same as multiplying by its reciprocal.
There are several ways to multiply fractions and whole numbers. Here's one.


21. Let x = the suggested selling price of the car. The first and second sentences tell you that of the suggested price = $21,000. So your equation is:
Divide both sides of the equation by

Dividing by a fraction is the same as multiplying by its reciprocal.
	x = $24,000
22. Let b = the number of bears in each packing crate. The first sentence tells you that the number of packing crates times the number of bears in each is equal to the total number of bears. Your equation is:	54b = 324
Divide both sides of the equation by 54.	54b ÷ 54 = 324 ÷ 54
	b = 324 ÷ 54
Divide.	b = 6
23. Let t = the number of turtle hatchlings born. The first sentence tells you that only 3% survive to adulthood. Three percent of the turtles born is 1,200. Your equation will be:	(3%)t = 1,200
The numerical equivalent of 3% is 0.03, so the equation becomes	0.03t = 1,200.
Divide both sides of the equation by 0.03.	0.03t ÷ 0.03 = 1,200 ÷ 0.03
	t = 1,200 ÷ 0.03
Divide.	t = 40,000
24. Let c = the number of acres he planted last year. 1.5 times c is 300.	1.5c = 300
Divide both sides of the equation by 1.5.	1.5c ÷ 1.5 = 300 ÷ 1.5
	c = 300 ÷ 1.5
Divide.	c = 200
25.Let d = her annual salary. Five percent of her salary equals her yearly bonus. Your equation will be:	(5%)d = $6,000
The numerical equivalent of 5% is 0.05, so the equation becomes	0.05d = 6,000.
Divide both sides of the equation by 0.05.	0.05d ÷ 0.05 = 6,000 ÷ 0.05
	d = 6,000 ÷ 0.05
Divide.	d = $120,000