• Verify a solution of an equation
  • Solve equations using the properties of equality
  • Solve equations that require simplification
  • Translate to an equation and solve
  • Translate and solve applications

Assignment

1-5-9s are the exercise numbers that end in either a 1, 5, or 9. In other words, do problems 1, 5, 9, 11, 15, 19, 21, 25, 29, …

Verify a Solution of an Equation

Equations are mathematical statements, specifically they are considered true statements. This means if part of the equation is unknown (represented by a variable), the goal is typically to find a solution, or a value that ensures the equation remains true.

To see if a solution works, you substitute it into the equation. Let’s check if $x=\frac{3}{2}$ is a solution to the equation ${4x-2=2x+1}$.

\[\begin{align} 4x - 2 &= 2x + 1 \\ 4\left(\frac{3}{2}\right) - 2 &= 2\left(\frac{3}{2}\right) + 1 \\ 6 - 2 &= 3 + 1 \\ 4 &= 4 \end{align}\]

Notice that when substituting I used parentheses. Handwriting can be misread, so this helps to avoid any confusion about what operation is taking place.

Solve Equations Using the Properties of Equality

In equations, variables can represent an unknown quantity. So, ${x+3=8}$ means that some number increased by $3$ is equal to $8$. You might be able to determine $x$ without much work, but let’s use the Subtraction Property of Equality.

Subtraction Property of Equality

For any numbers $a$, $b$, and $c$, \(\begin{align} &\text{If} & a &= b \\ &\text{then} & a-c&=b-c \end{align}\)

If you subtract one quantity from one side, then you need to do the same to the other for the equation to remain true. So, with our equation we can subtract $3$ from both sides to determine the value of $x$, the solution to the equation.

\[\begin{align} x + 3 &= 8 \\ x + 3 - 3 &= 8 - 3 \\ x &= 5 \end{align}\]

And this works with addition as well, since addition and subtraction are essentially the same operation.

Addition Property of Equality

For any numbers $a$, $b$, and $c$, \(\begin{align} &\text{If} & a &= b \\ &\text{then} & a+c&=b+c \end{align}\)

With multiplication and division, the goal is slightly different. Before, we were trying to get to $0$. With multiplication and division, now we want to get to $1$. Below, the original equation tells us what $4x$ is worth, but we want to know the value of just $1x$, or just $x$.

\[\begin{align} 4x &= 12 \\ \frac{4x}{4} &= \frac{12}{4} \\ x &= 3 \end{align}\]

We use the Multiplication and Division Properties of Equality to solve these types of equations.

Division Property of Equality

For any numbers $a$, $b$, and $c$, \(\begin{align} &\text{If} & a &= b \\ &\text{then} & \frac{a}{c}&=\frac{b}{c} \end{align}\)

Multiplication Property of Equality

For any numbers $a$, $b$, and $c$, \(\begin{align} &\text{If} & a &= b \\ &\text{then} & ac &= bc \end{align}\)

A useful thing to keep in mind when dealing when multiplying with fractions is reciprocals. The reciprocal of a fraction is the upside-down version of it. So, the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$. And if you multiply a fraction by its reciprocal, the result is always $1$.

\[\begin{align} \frac{2}{5} \cdot \frac{5}{2} = \frac{10}{10} = 1 \end{align}\]

This is useful for equations like this one.

\[\begin{align} \frac{3}{4}x &= 9 \\ \left(\frac{4}{3}\right) \left(\frac{3}{4}x\right) &= (9)\left(\frac{4}{3}\right) \\ x &= \frac{36}{3} \\ x &= 12 \end{align}\]

Solve Equations That Require Simplification

Before applying any of the above, it’s usually best to simplify both sides of the equation to make the problem easier to work with. Typically, this means just combining like terms and distributing.

\[\begin{align} -6(x − 2) − 15 &= 30 \\ -6x + 12 - 15 &= 30 \\ -6x - 3 &= 30 \\ -6x &= 33 \\ x &= -\frac{33}{6} \end{align}\]

And you will end up with fractions as answers. Simplify when possible, but feel free to leave them as improper fractions.