Solving Linear Equations With a Single Variable

Solving Linear Equations With a Single Variable

using properties of equality

Linear Equations

A linear equation is an equation whose graph will make a straight line. Linear equations do not include variables raised to a power or square roots of variables.

Multiple Select

Which of the equations shown below would be an example of a linear equation with a single variable? Remember that a linear equation with a single variable will not include any variables raised to a power or any roots of variables. These equations will also not include more than one variable, although the same variable can appear more than once. Check all that apply.

• $$a^2+25=50$$  

• $$4x-12=63$$  

• $$5\left(b-7\right)=8b+15$$  

• $$5c+d=45-a$$  

Steps for solving a linear equation with a single variable

1) Apply the Distributive Property if possible

2) Combine like terms on each side of the equation if necessary

3) Simplify by using properties of equality so that you're left with a variable term on one side and a constant on the other

4) Isolate the variable to find it's value

Step 1: Apply the Distributive Property if possible

If an equation has parentheses, you should be able to simplify it using the Distributive Property. The Distributive Property states that you can distribute multiplication across addition or subtraction. Some examples are shown to the right.

Multiple Choice

Which of the following equations would be equivalent to  $$4x+3=8\left(x+4\right)$$  after the Distributive Property has been applied?

• $$4x+7=12$$  

• $$4x+3=8x+32$$  

• $$4x+3=8+4x$$  

Multiple Choice

Which of the following equations would be equal to  $$5\left(4-y\right)=45-6y$$ after the Distributive Property has been applied? 

• $$20-y=39y$$  

• $$4-5y=45-6y$$  

• $$20-5y=45-6y$$  

Step 2: Combine like terms on each side of the equation if necessary

Like terms can be combined on each side of the equation to simplify. Remember like terms are terms that either have the same variable or no variable at all.

Multiple Choice

Which of the following equations would be equivalent to $$4x+6+8x=2x+18$$  after like terms are combined? Remember you can only combine like terms on one side of the equals sign.

• $$12x+6=2x+18$$  

• $$14x=12$$  

• $$10x+8=2x+18$$  

Multiple Choice

Which of the following equations would be equivalent to $$9y-14=5y-12-6$$  after like terms are combined? Remember you can only combine like terms on one side of the equals sign, and anything being subtracted must be treated as a negative value.

• $$14y-14=-18$$  

• $$9y-14=5y-18$$  

• $$9y-14=-7y-6$$  

Step 3: Simplify by using properties of equality

When you have variable terms on both sides of the equation, you can simplify by either adding or subtracting the same amount from both sides. You want to simplify the equation so that you have one variable term left on one side and a constant on the other.

Multiple Choice

Which of the following equations would be equal to $$12x+6=9x+15$$ after simplifying by using properties of equality?

• $$3x=21$$  

• $$21x=21$$  

• $$3x=9$$  

Multiple Choice

Which of the following equations would be equal to $$9x-30=4x+20$$ after simplifying by using properties of equality? Remember that anything being subtracted must be added to both sides in order to be canceled.

• $$5x=50$$  

• $$13x=10$$  

• $$13x=50$$  

Step 4: Isolate the variable to find its value

If you are able to get the variable on one side of the equation by itself, you will know its value. This is done by performing the opposite of what is being done to the variable and usually involves multiplication or division.

Multiple Choice

Solve the equation $$3y=27$$ by isolating the variable.

• $$y=3$$  

• $$y=9$$  

• $$y=87$$  

Multiple Choice

Solve the equation $$\frac{z}{-4}=12$$  by isolating the variable.

• $$z=8$$  

• $$z=-3$$  

• $$z=-48$$  

Now we will solve an equation from start to finish using all four steps

We will perform one step at a time

Multiple Choice

Step 1: Simplify the equation $$4\left(x+6\right)+2x=12x-18$$ by applying the Distributive Property. We are only performing one step at a time.

• $$4x+8x=12x-18$$  

• $$4x+24+2x=12x-18$$   

• $$8x+6=12x-18$$  

Multiple Choice

Step 2: Simplify the equation $$4x+24+2x=12x-18$$ by combining like terms. We are only performing one step at a time. 

• $$6x+24=12x-18$$  

• $$30x=12x-18$$  

• $$6x+24=-6x$$  

Multiple Choice

Step 3: Simplify the equation $$6x+24=12x-18$$ by using properties of equality. Remember to cancel terms by either adding or subtracting the same thing from both sides. 

• $$6=6x$$  

• $$18x=6$$  

• $$42=6x$$  

Multiple Choice

Step 4: Solve the equation $$42=6x$$ by isolating the variable. Remember to perform the opposite of what is being done to the variable on both sides. 

• $$x=12$$  

• $$x=7$$  

• $$x=252$$  

Here is a recap of the steps we took to solve this problem

Step 1: Apply the Distributive Property

Step 2: Combine Like Terms

Step 3: Simplify using properties of equality

Step 4: Isolate the variable

These steps an be used to solve ANY linear equation with a single variable.

Good luck with solving equations!