\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Solving Equations

Definitions


Welcome to the Math Escape Room! Today, you will embark on a journey to find the secret one-digit code. With your team, you must use your problem-solving skills to crack the code!
The code is:$$\triangle + 10 = 1 + 2 \times 6$$$$\triangle = \boxed{\phantom{\triangle}}$$You have only one opportunity.
  • Louis proposes to enter \(\boxed{8}\)
  • Su proposes to enter \(\boxed{3}\)
Which code will you use to enter?

We will test each proposed code to determine which one satisfies the given equation.
  • Test Louis's code: \(\textcolor{colordef}{\triangle = 8}\) $$ \begin{aligned} \textcolor{colordef}{(8)} + 10 &= 1 + 2 \times 6 \quad &&\text{(Substitute)} \\ 18 &= 1 + 12 \\ 18 &= 13 \quad &&\text{(False)} \end{aligned} $$
  • Test Su's code: \(\textcolor{colordef}{\triangle = 3}\) $$ \begin{aligned} \textcolor{colordef}{(3)} + 10 &= 1 + 2 \times 6 \quad &&\text{(Substitute)} \\ 13 &= 1 + 12 \\ 13 &= 13 \quad &&\text{(True)} \end{aligned} $$
Therefore, the correct code to enter is: \(\triangle = 3\).


Definition Solving an Equation
Solving an equation involves finding the values of the variable, called solutions, that make the equation true.
In this context, the variable is called the unknown.
We often use the letter \(x\) to represent the unknown.
Example
Show that a solution of \(3 + x = 5\) is \(x = 2\).

For \(\textcolor{colordef}{x = 2}\):$$\begin{aligned}3 + \textcolor{colordef}{(2)} &= 5 \quad &&\text{(substituting)} \\ 5 &= 5 \quad &&\text{(True)}\end{aligned}$$

Example
Show that \(x = 1\) is not a solution of \(3 + x = 5\).

For \(\textcolor{colordef}{x = 1}\):$$\begin{aligned}3 + \textcolor{colordef}{(1)} &= 5 \quad &&\text{(substituting)} \\ 4 &= 5 \quad &&\text{(False)}\end{aligned}$$

Solving by Trial and Error

Method Trial and Error
The trial and error method is a problem-solving strategy used to find a solution to an equation by testing different values for the unknown variable until the correct value is found.
Example
Consider the equation \(2x + 3 = 11\).
Use the trial and error method to find a solution.

  • Let's try \(\textcolor{colordef}{x = 2}\): $$ \begin{aligned} 2 \times \textcolor{colordef}{(2)} + 3 &= 11 \quad &&\text{(Substitute)} \\ 4 + 3 &= 11 \\ 7 &= 11 \quad &&\text{(False)} \end{aligned} $$
  • Let's try \(\textcolor{colordef}{x = 3}\): $$ \begin{aligned} 2 \times \textcolor{colordef}{(3)} + 3 &= 11 \quad &&\text{(Substitute)} \\ 6 + 3 &= 11 \\ 9 &= 11 \quad &&\text{(False)} \end{aligned} $$
  • Let's 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 \(x = 4\).