5.01 Solving equations
Introduction
When solving problems that involve missing or unknown values, we will often replace that value with a pronumeral in order to make an equation.
For example: Jill's age is equal to 5 more than twice the age of Jack. If Jill's age is 23, how old is Jack?
If we replace Jill's age with 23 and Jack's age with the pronumeral x, we can solve this question using the equation:23=5+2xTo find Jack's age, we can solve this equation to find the value of x.
Reverse operations
To solve equations using algebra, the most important rule to remember is that if we apply operations to one side of the equation, we must also apply it to the other.
When applying operations to equations, we always apply the same step to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.
Making sure to follow this rule, we can isolate the pronumeral in an equation by applying operations to both sides of the equation which reverse the operations applied to the pronumeral.
To do this, we need to know which operations are reversed by which.
| Operation | Reverse operation | Example |
|---|---|---|
| \text{Addition} | \text{Subtraction} | x+4-4=x |
| \text{Subtraction} | \text{Addition} | x-3+3=x |
| \text{Multiplication} | \text{Division} | y\times4\div4=y |
| \text{Division} | \text{Multiplication} | y\div 2\times 2=y |
Examples
Example 1
Solve the equation: 24+p=59
When applying operations to equations, we always apply the same operation to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.
Order of reverse operations
If we return to the case of finding Jack's age, we can see that there are two operations that have been applied to the pronumeral in the equation:23=5+2x
As we can see, to get 23 we multiplied x by 2 and then added 5. In order to isolate 23 we need to reverse these operations in the correct order. Which operation should we try to reverse first?
If we try to reverse the multiplication first, we would divide both sides by 2 to get:\dfrac{23}{2}=\dfrac{5+2x}{2}
This only makes the pronumeral less isolated and is not what we want.
When reversing operations, we apply them in the reverse of the usual order of operations.
When applying reverse operations to isolate a pronumeral, we apply them according to the order of reverse operations:
Addition and subtraction
Multiplication and division
Expressions inside brackets
Knowing this, we can now solve the equation to find Jack's age.
| \displaystyle 23 | \displaystyle = | \displaystyle 5+2x | |
| \displaystyle 18 | \displaystyle = | \displaystyle 2x | Reverse the addition |
| \displaystyle 9 | \displaystyle = | \displaystyle x | Reverse the multiplication |
As such, we find that Jack's age is 9.
Examples
Example 2
Consider the equation 2(p+6)=-8
Which pair of operations will make p the subject of the equation?\begin{array}{ccccc} & \text{Step } 1 & & \text{Step } 2 & \\ 2(p+6) & \longrightarrow & p+6 & \longrightarrow & p \end{array}
Apply these operations to the right-hand side of the equation as well.\begin{array}{ccccc} & \text{Divide by } 2 & & \text{Subtract } 6 & \\ -8 & \longrightarrow & ⬚ & \longrightarrow & ⬚ \end{array}
Using your answer from part (b), what value of p will make the equation 2(p+6)=-8 true?
Example 3
Solve the equation \dfrac{u+7}{2}=5.
When applying reverse operations to isolate a pronumeral, we apply them according to the order of reverse operations:
Addition and subtraction
Multiplication and division
Expressions inside brackets
Move the pronumeral first
With the skills we now have, we can reverse as many operations as we need to isolate our pronumeral. But what if we want to solve equations like 4-x=17 or \dfrac{5}{x}=20?
In both of these cases we can see that the pronumeral is actually part of the operation being applied, so we can't isolate it so easily.
However, if we reverse the operation containing the pronumeral, we can move it so that the equation can be solved using the skills we just learned.
Examples
Example 4
Solve the equation \dfrac{-44}{p}=11.
If the pronumeral is actually part of the operation being applied, we can reverse the operation containing the pronumeral.
For example, in the following equation we are dividing by x: \dfrac{5}{x}=2 so we can use the reverse operation and multiply both sides by x: \begin{aligned} \dfrac{5}{x}\times x & =2 \times x \\ 5 &=2x \end{aligned}