3.5 Solve Equations Using Integers; The Division Property of Equality

Solving Equations with Integers

When you have an equation like $3x = -12$, your job is to find the integer that makes both sides equal. This section covers how to check whether a value is a solution and how to use the division property of equality to solve equations where the variable is multiplied by a number.

Checking Integer Solutions

To check whether a given integer is a solution, substitute it into the equation and simplify each side. If both sides are equal, that integer is a solution.

1. Replace the variable with the integer you're testing.
2. Simplify each side using order of operations (PEMDAS).
3. Compare the two sides.

Example: Is $x = -3$ a solution to $4x + 5 = -7$?

• Left side: $4(-3) + 5 = -12 + 5 = -7$
• Right side: $-7$
• Both sides equal $-7$, so yes, $x = -3$ is a solution.

If the two sides don't match, that integer is not a solution.

Properties for Solving Equations

You've already seen two properties that help isolate a variable:

• Addition Property of Equality: If $a = b$, then $a + c = b + c$
• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$

These let you add or subtract the same number from both sides to move a constant away from the variable. Whatever you do to one side, you must do to the other to keep the equation balanced.

The Division Property of Equality

What the Property Says

The division property of equality states that you can divide both sides of an equation by the same non-zero number and the equation stays true.

You use this property when a variable is being multiplied by a number (its coefficient) and you need to undo that multiplication. Dividing by zero is never allowed, which is why the rule requires $c \neq 0$.

Solving Equations with Division

When the variable has a coefficient, divide both sides by that coefficient to isolate the variable.

Example: Solve $-5x = 40$

1. The coefficient of $x$ is $-5$.
2. Divide both sides by $-5$: $\frac{-5x}{-5} = \frac{40}{-5}$
3. Simplify: $x = -8$
4. Check: $-5(-8) = 40$ ✓

Example with a negative result: Solve $7x = -21$

1. Divide both sides by $7$: $\frac{7x}{7} = \frac{-21}{7}$
2. Simplify: $x = -3$
3. Check: $7(-3) = -21$ ✓

Pay attention to integer sign rules when dividing. A negative divided by a negative gives a positive. A positive divided by a negative (or vice versa) gives a negative.

Translating Word Problems into Equations

Many problems give you a situation in words, and you need to build an equation before solving.

1. Identify the unknown and assign it a variable (e.g., let $n$ = the number).

2. Translate key phrases into math operations:

   • "product of" → multiplication
   • "quotient of" → division
   • "sum of" → addition
   • "difference of" → subtraction

3. Write the equation using the information given.

4. Solve using the appropriate property of equality.

5. Check your answer in the context of the problem. Does it make sense? (For example, a count of people can't be negative or a fraction.)

Example: "The product of $-6$ and a number is $54$. Find the number."

• Equation: $-6n = 54$
• Divide both sides by $-6$: $n = \frac{54}{-6} = -9$
• Check: $-6(-9) = 54$ ✓

Types of Equations in This Section

All the equations you'll see here are linear equations, meaning the variable has an exponent of 1 (like $3x$, not $x^2$). You may sometimes need to simplify one side first using the distributive property before applying the division property, but the core process stays the same: isolate the variable, then solve.