General Methods for Solving Equations

Solving equations is the process of determining the value(s) of the unknown variables that make an equation true. 

Balancing equations consists of isolating a variable to one side of the equation, using the rules for transforming equations.

Find the value of |x| in the following equation: |2x + 5 = x + 7| 

1. To eliminate the algebraic term |x| from the right-hand side, subtract it from both sides of the equation. ||\begin{align}2x + 5 \color{red}{- x} &= x + 7 \color{red}{- x} \\ x + 5 &= 7 \end{align}||

2. To isolate |x| on the left-hand side, subtract |5| from both sides of the equation. ||\begin{align} x + 5 \color{red}{- 5} &= 7 \color{red}{- 5} \\ x &= 2 \end{align}||

Conclusion: |x = 2.|

Find the value of |x| in the following equation: |\displaystyle \frac{3x}{4} - 2{.}5 = 2{.}3|

1. To isolate the term |\displaystyle \frac{3x}{4}| in the left-hand side, add |2{.}5| to both sides of the equation. ||\begin{align} \frac{3x}{4} - 2{.}5 \color{red}{+ 2{.}5} &= 2{.}3 \color{red}{+ 2{.}5} \\ \frac{3x}{4} &= 4{.}8 \end{align}||

2. To isolate the term |3x| on the left-hand side, multiply both sides of the equation by |4|. ||\begin{align} \frac{3x}{4}\color{red}{\times 4} &= 4{.}8\color{red}{\times 4} \\ 3x &= 19{.}2 \end{align}||

3. To isolate |x| on the left-hand side, divide both sides of the equation by |3|. ||\begin{align} \color{red}{\frac{\color{black}{3x}}{3}} &= \color{red}{\frac{\color{black}{19{.}2}}{3}} \\ x &= 6{.}4 \end{align}||

Conclusion: |x = 6{.}4.|

The inverse operations method consists of isolating an unknown variable by applying the inverse of each operation to both sides of the equation.

Find the value of |x| in the following equation: |\displaystyle \frac{2x}{3} - 16 = -6|

1.  Transform the equation’s operations into inverse operations. Make sure to also reverse the order in which the operations are usually completed. ||\begin{align} &x \to \times 2 \to \div 3 \to - 16\ = -6 \\ &x =\: \div 2 \leftarrow \times 3 \leftarrow + 16 \leftarrow -6 \end{align}||

2. The operations are completed from right to left. ||\begin{align} x &= \div 2 \leftarrow \times 3 \leftarrow \color{red}{+ 16 \leftarrow -6}\\ x &= \div 2 \leftarrow \color{red}{\times 3 \leftarrow + 10}\\ x &= \color{red}{\div 2 \leftarrow 30} \\ x &= 15 \end{align}||

Conclusion: |x = 15.|

The hidden terms method, also called the cover up method, consists of hiding an algebraic term in order to find the value of the hidden term afterward.

Find the value of |x| in the following equation: |\displaystyle \frac{5x}{3} - 12 = 8.|

1. Look for the value of |\displaystyle \frac{5x}{3}.|

   Cover the term |\displaystyle \frac{5x}{3}| in the equation. ||\begin{align} \color{red}{?} - 12 &= 8\\ \color{red}{20} - 12 &= 8 \end{align}|| Observe that |\displaystyle \frac{5x}{3} = 20.|

2. Next, look for the value of |5x.|

   Hide the term |5x| in the equation. ||\begin{align} \frac{\color{red}{?}}{3} = 20\\ \frac{\color{red}{60}}{3} = 20 \end{align}|| Observe that |5x = 60.|

3. Look for the value of |x.|

   Hide the term |x| in the equation. ||\begin{align} 5 \times\ \color{red}{?}\ &= 60 \\ 5\times \color{red}{12} &= 60 \end{align}|| Observe that |x = 12.|

Conclusion: |x = 12.|

The trial-and-error method involves trying out different possible values for a variable and checking to see if they are solutions for the equation.

Find the value of |x| in the following equation: |\displaystyle \frac{x}{2} + 6 = 10.|

1^st test: Replace |x| with |2.| ||\begin{align} \frac{\color{red}{2}}{2} + 6 &\overset{?}{=} 10 \\ 1 + 6 &\overset{?}{=} 10 \\ 7 &\neq 10 \end{align}|| The two sides of the equation are not equal, because |7 < 10.| Observe that the solution is greater than |2.|

2^nd test: Replace |x| with |10.| ||\begin{align} \frac{\color{red}{10}}{2} + 6 &\overset{?}{=} 10 \\ 5 + 6 &\overset{?}{=} 10 \\ 11 &\neq 10 \end{align}|| The two sides of the equation are not equal to each other, because |11>10.| Observe that the solution is less than |10.|

3^rd test: Replace |x| with |8.| ||\begin{align} \frac{\color{red}{8}}{2} + 6 &\overset{?}{=} 10 \\ 4 + 6 &\overset{?}{=} 10 \\ 10 &= 10 \end{align}||

The two sides of the equation are equal to each other. Conclusion: the solution is |x = 8.|

Verifying an equation’s solution is a process used to check the accuracy of the value of the variable found.

The solution |x = 12| was obtained in the example on the cover-up method seen above. In order to check if this answer makes the initial equation true, simply replace the variable with the value of the solution. 

||\begin{align} \frac{5x}{3} -12 &= 8 \\ \frac{5 \times \color{red}{12}}{3} -12 &\overset{?}{=} 8 \\ \frac{60}{3} -12 &\overset{?}{=} 8 \\ 20 -12 &\overset{?}{=} 8 \\ 8 &= 8 \end{align}||

Since both sides of the equation are equal, the conclusion that |x=12| is correct.