# Solving Linear Equations

Goals

- Solve linear equations using addition and subtraction.
- Solve linear equations using multiplication and division.
- Use linear equations to solve real-life problems.
- Solve multi-step linear equations using inverse operations.
- Use multi-step linear equations to solve real-life problems.
- Use unit analysis to model real-life problems.
- Solve linear equations that have variables on both sides.
- Identify special solutions of linear equations.
- Solve absolute value equations.
- Solve equations involving two absolute values.
- Identify special solutions of absolute value equations.
- Rewrite literal equations.
- Rewrite and use formulas for area.
- Rewrite and use other common formulas.

Vocabulary

- A
is an unproven statement about a general mathematical concept.*conjecture* - A proven statement about a general mathematical concept is a
or a*rule*.*theorem* - An
is a statement that two expressions are equal.*equation* - A
is an equation that can be written in the form*linear equation in one variable**ax*+*b*= 0, where*a*and*b*are constants and*a*≠ 0. - A
of an equation is a value that makes the equation true.*solution* are two operations that undo each other, such as addition and subtraction.*Inverse operations*are equations that have the same solution(s).*Equivalent equations*- An equation that is true for all values of the variable is an
and has infinitely many solutions.*identity* - An
is an equation that contains an absolute value expression.*absolute value equation* - An
is an apparent solution that must be rejected because it does not satisfy the original equation.*extraneous solution* - An equation that has two or more variables is called a
.*literal equation* - A
shows how one variable is related to one or more other variables.*formula*

Concepts

__Addition Property of Equality__

- Adding the same number to each side of an equation produces an equivalent equation.
- If
*a*=*b*, then*a*+*c*=*b*+*c*.

__ __

__Subtraction Property of Equality__

- Subtracting the same number from each side of an equation produces an equivalent equation.
- If
*a*=*b*, then*a*−*c*=*b*−*c*.

__Multiplication Property of Equality__

- Multiplying each side of an equation by the same nonzero number produces an equivalent equation.
- If
*a*=*b*, then*a*⋅*c*=*b*⋅*c*,*c*≠

__Division Property of Equality__

- Dividing each side of an equation by the same nonzero number produces an equivalent equation.
- If
*a*=*b*, then*a*÷*c*=*b*÷*c*,*c*≠

__Solving Multi-Step Equations__

- To solve a multi-step equation, simplify each side of the equation, if necessary.
- Then use inverse operations to isolate the variable.

__Solving Equations with Variables on Both Sides__

- To solve an equation with variables on both sides, simplify one or both sides of the equation, if necessary.
- Then use inverse operations to collect the variable terms on one side, collect the constant terms on the other side, and isolate the variable.

__Special Solutions of Linear Equations__

- Equations do not always have one solution.
- An equation that is true for all values of the variable is an identity and has infinitely many solutions.
- An equation that is not true for any value of the variable has no solution.

__Four-Step Approach to Problem Solving__

__ __

**Understand the Problem**

What is the unknown? What information is being given? What is being asked?

**Make a Plan**

This plan might involve one or more common problem-solving strategies.

**Solve the Problem**

Carry out your plan. Check that each step is correct.

**Look Back**

Examine your solution. Check that your solution makes sense in the original statement of the problem.

__ __

__Properties of Absolute Value__

Let *a *and *b *be real numbers. Then the following properties are true.

__Solving Equations with Two Absolute Values__

To solve ∣ *ax *+ *b *∣ = ∣ *cx *+ *d *∣ , solve the related linear equations *ax *+ *b *= *cx *+ *d *or *ax *+ *b *= −(*cx *+ *d*).

__Common Problem-Solving Strategies__

- Use a verbal model.
- Guess, check, and revise.
- Draw a diagram.
- Sketch a graph or number line.
- Write an equation.
- Make a table.
- Look for a pattern.
- Make a list.
- Work backward.
- Break the problem into parts.

__Steps for Solving Linear Equations__

Here are several steps you can use to solve a linear equation. Depending on the equation, you may not need to use some steps.

** **

**Step 1 **Use the Distributive Property to remove any grouping symbols.

**Step 2 **Simplify the expression on each side of the equation.

**Step 3 **Collect the variable terms on one side of the equation and the constant terms on the other side.

**Step 4 **Isolate the variable.

**Step 5 **Check your solution.

__Solving Absolute Value Equations__

- To solve ∣
*ax*+*b*∣ =*c*when*c*≥ 0, solve the related linear equations*ax*+*b*=*c or*

*ax *+ *b *= − *c*.

- When
*c*< 0, the absolute value equation

∣ *ax *+ *b *∣ = *c *has no solution because absolute value always indicates a number that is not negative.

__Common Formulas__

**Temperature**

*F *= degrees Fahrenheit

*C *= degrees Celsius

*C *= (*F *− 32)

**Simple Interest**

*I *= interest

*P *= principal

*r *= annual interest rate (decimal form)

*t *= time (years)

*I *= *Prt*

* *

**Distance **

*d *= distance traveled, *r *= rate, *t *= time

*d *= *rt*

Activities

Challenge Yourself …

Which of the following equations have only one solution? Which have two solutions? Which have no solution? Which have infinitely many solutions?