Solving Equations

What is an Equation?

Welcome to the Math Escape Room! Your team must solve a final puzzle to find the secret one-digit code (a number from $0$ to $9$) and escape.
The clue is an equation where the code is represented by the symbol $\triangle$:$$\triangle + 10 = 1 + 2 \times 6$$You only have one chance to enter the correct digit. Two of your teammates suggest different answers:

Louis proposes entering $\boxed{8}$.
Su proposes entering $\boxed{3}$.

Who is correct, and what is the secret code?

Answer

To find the correct code, we test each suggestion to see which value for $\triangle$ makes the equation a true statement.
First, simplify the right-hand side of the equation. Remember that we do the multiplication first:$$1 + 2 \times 6 = 1 + 12 = 13.$$So the equation becomes:$$\triangle + 10 = 13.$$

Test Louis's code: Substitute $\textcolor{colordef}{\triangle = 8}$ into the equation. $$ \begin{aligned} \textcolor{colordef}{8} + 10 &= 18 \\ 18 &= 13 &&\text{(False)} \end{aligned} $$ Louis's code is incorrect.
Test Su's code: Substitute $\textcolor{colordef}{\triangle = 3}$ into the equation. $$ \begin{aligned} \textcolor{colordef}{3} + 10 &= 13 \\ 13 &= 13 &&\text{(True)} \end{aligned} $$ Su's code is correct.

The secret code is $\triangle = 3$ because this is the value that makes the equation true.

Definition Equation and Solution

An equation is a mathematical statement that says two expressions are equal. It often contains a variable (or unknown), which is a symbol (like $x$ or $\triangle$) representing a number we do not yet know.
Solving an equation means finding all value(s) of the variable that make the equation a true statement. Each of these values is called a solution of the equation.

Example

Show that $x = 2$ is a solution to the equation $3 + x = 5$.

Answer

We substitute $\textcolor{colordef}{x = 2}$ into the equation and check if the left-hand side equals the right-hand side:$$\begin{aligned}3 + \textcolor{colordef}{(2)} &= 5 \\ 5 &= 5 \quad &&\text{(This is a true statement.)}\end{aligned}$$Since the statement is true, $x = 2$ is a solution.

Example

Show that $x = 1$ is not a solution to $3 + x = 5$.

Answer

We substitute $\textcolor{colordef}{x = 1}$ into the equation:$$\begin{aligned}3 + \textcolor{colordef}{(1)} &= 5 \\ 4 &= 5 &&\text{(This is a false statement.)}\end{aligned}$$Since the statement is false, $x = 1$ is not a solution.

Solving by Inspection and Trial-and-Error

Method Trial and Error

Trial and error is a basic problem-solving method where we test different values for the variable until we find one that makes the equation true. For each value, we substitute it into the equation and check whether the left-hand side equals the right-hand side.
For simple equations, we can sometimes see the solution just by looking at the equation. This is called solving by inspection.

Example

Consider the equation $2x + 3 = 11$.
Use the trial and error method to find the solution.

Answer

We test different integer values for $x$ to see which one makes the equation true.

Try $\textcolor{colordef}{x = 2}$: $$ \begin{aligned} 2 \times \textcolor{colordef}{(2)} + 3 &= 11 \quad &&\text{(Substitute)} \\ 4 + 3 &= 11 \\ 7 &= 11 \quad &&\text{(False)} \end{aligned} $$
Try $\textcolor{colordef}{x = 3}$: $$ \begin{aligned} 2 \times \textcolor{colordef}{(3)} + 3 &= 11 \quad &&\text{(Substitute)} \\ 6 + 3 &= 11 \\ 9 &= 11 \quad &&\text{(False)} \end{aligned} $$
Try $\textcolor{colordef}{x = 4}$: $$ \begin{aligned} 2 \times \textcolor{colordef}{(4)} + 3 &= 11 \quad &&\text{(Substitute)} \\ 8 + 3 &= 11 \\ 11 &= 11 \quad &&\text{(True)} \end{aligned} $$

Therefore, a solution to the equation $2x + 3 = 11$ is $x = 4$.