MA007 Study Guide

2a. Determine whether a given real number is a solution to an equation

• How do you know if a value is a solution to an equation?

A value is considered a solution to an equation if you can replace the variable with that value and the outcome is true. To check if a value is a solution, first, substitute the value in place of the variable. Then, simplify the expressions on both sides of the equation. Finally, determine if the final statement is true. 

For example, determine if \(x = 3\) is a solution to the equation \(2x+3=5x-2\). First, substitute 3 in place of \(x\) on both sides of the equation \(2(3)+3=5(3)-2\). Next, simplify both sides of the equation: \(6+3=15-2\). Finally, determine if the final statement is true: \(9=13\) is not true; therefore, 3 is not a solution to this equation. 

To review, see:

• Testing Solutions to Equations

2b. Simplify equations using addition and multiplication properties

• How do you simplify equations using addition properties?
• How do you simplify equations using multiplication properties?

When solving equations, we can use some basic properties to find the value of the variable. When a constant is added or subtracted from your variable, you can isolate the variable by doing the opposite operation with the same value on both sides of the equation. For example, to solve the equation \(x+4=7\), notice that 4 is currently being added to our variable, \(x\). To solve this equation, simply subtract 4 from both sides. This will give the answer \(x=3\).

Similarly, if the variable is being multiplied or divided by a number, you can again perform the opposite operation with the same number on both sides of the equation to solve for the variable. For example, when solving the equation \(\dfrac{x}{4}=3\), notice that the variable x is divided by 4. Multiply both sides of the equation by 4 to solve for \(x\). \(\dfrac{x}{4} \times 4=3 \times 4\). Therefore, \(x = 12\).

To review, see:

• The Subtraction and Addition Properties of Equality
• The Division and Multiplication Properties of Equality

2c. Find the solution of a given linear equation with one variable

• How do you solve equations using addition and subtraction properties?
• How do you solve equations using multiplication and division properties?
• How do you solve 2-step equations?
• How do you solve equations with variables on both sides?
• How do you solve equations with parentheses?

When solving an equation with multiple steps, always begin by eliminating any addition or subtraction first before addressing any multiplication or division. For example, we can use addition/subtraction properties in the equation \(3x - 2 = 7\). Begin by adding 2 to both sides since there is \(a -2\) already there. This yields \(3x - 2 + 2 = 7 + 2\), which simplifies to \(3x = 9\). Now, we can use multiplication/division properties to divide both sides by 3, giving us a final answer of \(x = 3\). 

If there is an equation with numbers and variables on both sides, first, you must collect all terms with the variable on one side and all terms without variables (the constants) on the other side of the equal sign. Then, solve the equation as described above. For example, \(3x+1=2x+5\) it would be solved by subtracting \(2x\) from both sides to get \(3x-2x+1=2x-2x+5\). This gives us \(x+1=5\). Then, subtract 1 from both sides to get \(x+1-1=5-1\), which gives the final answer \(x=4\).

If the equation has parentheses, first use the order of operations or the distributive property to remove the parentheses, then solve as described above.

To review, see:

• The Subtraction and Addition Properties of Equality Change
• The Division and Multiplication Properties of Equality
• Equations with Variables on Both Sides
• Solving Equations with the Distributive Property

2d. Determine the number of solutions of a given linear equation in one variable

• Can a linear equation in one variable have no solutions?
• Can a linear equation in one variable have more than one solution?

In certain instances, equations have solutions that do not end with the variable equal to a single specific value, as we saw in \(2b\) and \(2c\). When solving an equation, if all the variables cancel, we end up with a statement with constants on both sides, like \(3x+5=3x-1\). When we solve this equation, we end up with \(5= -1\). Since this is false, this equation has no solutions. 

On the other hand, if the final solution is a true statement like \(3=3\), then we say the equation has infinitely many solutions or all real numbers because any value for the variable will still simplify to \(3=3\), which is true.

To review, see:

• Solving Linear Equations with One Variable

2e. Solve a literal equation for the given variable

• How do you solve an equation for a specific variable?

Sometimes, equations have more than one variable, and instead of finding a value for a specific variable, we need to simply solve the equation for a variable. For example, if we solve the equation \(3v+s=t-2\) for the variable \(s\), we simply use the same skills we have learned for solving equations by subtracting \(3v\) from both sides, giving us an answer of \(s=t-2-3v\). 

To review, see:

• Solving for One Variable

Unit 2 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• all real numbers
• infinitely many solutions
• no solutions
• opposite operation
• parentheses
• solution
• statement