When we have an equation in one variable, there are many different ways to solve it. We generally want to make moves that get us closer to an equation that clearly shows the value that makes the equation true.

For example, $x=5$ or $t = \frac{7}{3}$ show that 5 and $\frac{7}{3}$ are solutions. Because there are many ways to do this, it helps to choose moves that leave fewer terms or factors.

If we have an equation like $3t + 5 = 7$, adding -5 to each side will leave us with fewer terms. The equation then becomes $3t = 2$.

Dividing each side of this equation by 3 results in the equivalent equation $t = \frac{2}{3}$, which is the solution.

Or, if we have an equation like $4(5 - a) = 12$, dividing each side by 4 will leave us with fewer factors on the left. The equation then becomes $5-a = 3$.

Here is a list of valid moves that can help create equivalent equations that move toward a solution:

1. Use the distributive property so that all the expressions no longer have parentheses.
2. Collect like terms on each side of the equation.
3. Add or subtract an expression on each side so that there is a variable on just one side.
4. Add or subtract an expression on each side so that there is just a number on the side without the variable.
5. Multiply or divide by a number on each side so that the variable on one side of the equation has a coefficient of 1.

For example, suppose we want to solve $9-2b + 6 =\text-3(b+5) + 4b$.

\begin{aligned} \text{Use the distributive property}&&9 - 2b + 6 &= \text-3b - 15 + 4b\\ \text{Combine like terms}&&15 - 2b &= b - 15\\ \text{Add \(2b to each side}&&15 &= 3b - 15\\ \text{Add 15 to each side}&&30 &= 3b\\ \text{Divide each side by 3}&&10 &= b\\ \end{align}\)

From lots of experience, we learn when to use different valid moves that help solve an equation.