In this explainer, we will learn how to solve multistep equations over the rational numbers.
Let’s start by recalling what we mean by “an equation” in mathematics. An equation is formed when two expressions of equal value are connected by an equals sign. For example,
Here, we know that the value of the expression is equal to 27. Equations are a powerful tool in mathematics; we often represent real-world situations using equations and then solve these to find unknown values.
We can think of an equation as a balance between two things. Much like with a set of balance scales, to keep the scales in balance, we have to add or remove equal masses from both sides of the scales. The same is true for an equation. If we apply an operation to one side of the equation, we must also apply the same operation to the other side to ensure that the values of the two sides remain equal.
Consider the equation that we mentioned earlier,
We could, for example, add 3 to each side of the equation to get which simplifies to
This represents an equivalent equation, as we have performed the same operation (in this case, addition) to each side, but this has not gotten us any closer to solving the equation. To solve an equation, we need to perform operations to both sides of the equation until we can write it in the form . So, for the equation , a sensible first step would be to subtract 7 from each side to get which simplifies to
Now, we know that 4 times something is equal to 20, so to find that something, we divide both sides by 4. We find that
An interesting note here is that although we could have divided the equation through by 4 first, this would have complicated our working out, as we would have had to divide all terms on each side by 4, which would have resulted in the equation
This is still equivalent, but it has introduced fractions, which are more difficult to work with. Therefore, when solving equations, we also consider the order of the operations that we apply to each side. This is particularly important when working with multistep equations.
What do we mean by “multistep equation”? Linear equations are often classified by the number of “steps” that are required to solve them. For example, an equation such as would often be referred to as a “one-step” equation, as we could find the solution by subtracting 8 from both sides. Similarly, an equation such as the one that we have just solved, is often referred to as a two-step equation, as two steps are required to solve it, namely subtracting 7 and dividing through by 4.
Multistep equations are a little more broadly defined, but intuitively, they are equations that require multiple steps to solve. This includes equations that contain an unknown variable more than once, including on each side of the equals sign. Examples of multistep equations include and
In both cases, the additional steps involve simplifying the equations down to two-step equations.
Our strategy when solving multistep equations should always be to apply operations to each side of the equation that simplify the equation into a form that is easier to work with.
A final noteworthy point when solving equations is that when we have solved an equation, we can always check our answer by substituting our solution back into the original equation. Recall again our original two-step equation, . we found that the solution to this was . If we substitute this back into our equation, we get which simplifies to
As the left-hand side is equal to the right-hand side, we know that our solution is correct.
Let’s now see some examples of solving equations by isolating the variable on one side of the equation.
Example 1: Solving a Linear Equation to Find a Rational Solution
Find the value of if .
We begin by recalling that we want to isolate on one side of the equation. We can do this by subtracting from both sides of the equation since this will remove the addition of from the left-hand side. We have
We then recall that to subtract two fractions, we need their denominators to be equal. We can note that 2 and 5 share no common factors, so their lowest common multiple is their product: . Thus, we will rewrite both fractions to have a denominator of 10. We get
Now that the denominators are equal, we can subtract the numerators to find the value of . This gives us
In our next example, we will look at how to solve an equation with an unknown and a constant on each side, a multistep equation.
Example 2: Solving a Multistep Linear Equation to Find a Rational Solution
Find the value of if .
To solve this equation, we need to isolate the variable on either the left- or right-hand side by applying the same operations to each side of the equation.
We can do this by either adding to each side of the equation or subtracting from each side. Generally, it will simplify our working if we identify the -term with the smallest coefficient, so we will add to each side. This gives us
We recall that we can add the -terms by collecting like terms and simplifying. This yields
Now we subtract 2 from both sides of the equation to get and divide both sides of the equation by 10 to get
Finally, we see that 4 and 10 share a factor of 2. Therefore, we can simplify as follows:
It is worth noting that we can check this answer by substituting into both sides of the original equation and checking they are equal. This is generally good practice, but it is not an essential step in our calculation, provided we have been accurate in our original calculation.
Substituting into the left-hand side of the equation yields
Substituting into the right-hand side of the equation yields
Since both sides of the equation are equal when , we have confirmed that this is a solution to the equation.
In our next example, we will look at how to solve an equation that contains expressions between parentheses.
Example 3: Solving a Multistep Linear Equation to Find a Rational Solution
Find the value of if .
To solve an equation, we need to isolate the variable on one side of the equation by applying the same operations to both sides of the equation.
Before applying any operations to the equation, we can start by simplifying the left-hand side: we distribute 2 over the first set of parentheses and negative 1 over the second set of parentheses. Remember that to distribute over the parentheses, we multiply each term in the parentheses by the multiplier. We get
We can simplify the left-hand side of the equation by combining like terms:
We can now rearrange the equation such that all of the -terms are on the same side of the equation. We add to both sides of the equation to get
Now, we subtract 5 from each side of the equation:
Finally, we divide the equation through by 5 to get
As we mentioned in the introduction, equations are often used as a useful tool in mathematics to solve real-world problems.
For example, if we are told that a rectangular field has a perimeter of 42 m and that its length is 3 m longer than its width , then we can note that the length of the field is .
The perimeter of a rectangle is the sum of twice its side lengths:
We can distribute 2 over the parentheses on the left-hand side of the equation to get
Simplifying then gives
Subtracting 6 from both sides yields
Dividing both sides of the equation by 4 then gives us
We can check this answer in two ways. We can sketch a rectangle with a width of 9 m and a length of and check its perimeter is 42 m.
Alternatively, we can substitute into our original equation to check that both sides of the equation are equal. We have
Sometimes in mathematics, we might be faced with a real-world scenario that we can represent using a linear equation and subsequently solve. We will demonstrate this in our next example.
Example 4: Forming and Solving a Linear Equation for a Real-World Problem
The sum of the weights of three people is 154 kg. If the first person is 2 kg heavier than the second person and the second person is 1 kg lighter than the third person, find each of their weights.
To answer this question, we need to form and solve an equation using the given information. We can label the weight of each person in kilograms using a different variable, say , , and .
Let’s now write each piece of information as an equation. First, the sum of the weights of the three people is 154 kg, so
Let’s consider that the first person is 2 kg heavier than the second person. This means that if we add 2 to the second person’s weight, we must get the weight of the first person. So,
Similarly, we are told that the second person is 1 kg lighter than the third person, so . must be the same as :
We can rearrange this equation so that is written in terms of . We add 1 to both sides of the equation to get
We now have expressions for the weights of the first person and the third person in terms of . We can substitute these expressions into the equation for the sum of the weights to get
Collecting like terms on the left-hand side of the equation yields
Subtracting 3 from both sides of the equation gives
Dividing both sides of the equation by 3 then gives
We can substitute this value of into our equations for and to find the weights of the other two people. We have
Hence, the weight of the first person is kg, that of the second is kg, and that of the third is kg.
In our final example, we will look at a problem involving the interior angles of a triangle. Much like with the previous example, we will ultimately solve the problem by forming and then solving an equation, but to do this, we will need to apply our knowledge about the sum of the interior angles of a triangle.
Example 5: Forming and Solving a Linear Equation for a Real-World Problem
In the following figure, find the measure of all the angles in triangle .
We begin by recalling that the sum of the measures of the interior angles in a triangle is equal to . Since we are told the measure of each angle, we can form an equation by finding their sum. We have
We can simply write this equation without the degree symbol for simplicity:
We can combine like terms by adding their coefficients. This gives
We can then divide both sides of the equation through by 7 to solve for :
We can then substitute this value for into each of the expressions for the measures of the angles.
For angle , we get
For angle , we get
For angle , we get
Let’s finish by recapping some of the important points from this explainer.
- We can solve equations by applying the same operations to both sides of the equation with the aim of isolating the variable on one side of the equation.
- A multistep equation is an equation in which multiple steps are needed to isolate the variable.
- It is important to check the order of operations when forming equations from a real-world scenario and also when solving them.