What is an Equation?
An equation is a mathematical statement that connects two equal expressions with an “equal to” sign. 3 x + 5 = 11 , y -2 = 3 , and 5z -6 = 9 are some examples of equations. In every true equation, the left-hand side will be equal to the right-hand side (LHS = RHS). The unknown values like x, y, and z found in these equations are known as variables. We can find the values of variables of an equation by rearranging the terms using some special properties.
Consider the equation 3 x + 5 = 11.
Here, 3 x + 5 and 11 are the expressions of the equation. 3x and 5 are the terms of the expression on the left hand side, and 11 is the only term in the expression on the right hand side. x is a variable, 3 is its coefficient, and 5 and 11 are constants. The plus sign ( + ) is an operator.
What is a Multi-Step Equation?
Some equations can be solved in two steps. On the other hand, we need to perform multiple operations like addition, subtraction, multiplication, and division to solve certain equations. Such equations are known as multi-step equations.
Properties of Equalities
We can use properties of equations to solve multi-step equations easily.
Addition property
The addition property of equality states that adding the same quantity on both sides of an equation gives us another equivalent equation.
That is, if a = b , then a + c = b + c
Subtraction property
According to the subtraction property of equality, if we subtract the same quantity from both sides of an equation, we get another equivalent equation.
If a = b , then a – c = b – c
Multiplication property
The multiplication property of equality states that an equation can be multiplied by the same quantity on both sides without affecting the solution of the equation.
If a = b , then a x c = b x c
Division property
The division property of equality states that an equation can be divided by the same quantity on both sides without affecting the solution of the equation.
If a = b, then a ÷ c = b ÷ c
How to Solve Multi-Step Equations
The main objective behind solving an equation is to find the value of an unknown variable. All equations can be solved by simplifying the terms on both sides and isolating the variable to find its solution. To isolate the variable, we need to use inverse math operations on both sides of the equation. The general procedure for solving multi-step equations is as follows:
We use the distributive property to expand the terms inside the parenthesis. Then, we combine the like terms to simplify both sides of the equation. Finally, we isolate the variable using inverse operations to arrive at the solution.
For example, let’s solve the equation 3 ( 5x + 2 ) = 36.
3 x ( 5x + 2 ) = 36 Write the equation
3 x 5x + 3 x 2 = 36 Expand the parentheses
15x + 6 = 36 Simplify
15x + 6 – 6 = 36 -6 Subtraction property of equality
15x = 30 Subtract
\(\frac{15x}{15}\) = \(\frac{30}{15}\) Division property of equality
x = 2 Simplify
So, the value of x in this equation is 2.
Solved Examples on Multi Step Equations
Example 1: Solve 18x – 8x + 20 = 70.
Solution:
18x – 8x + 20 = 70 Write the equation
10x + 20 = 70 Combine like terms
-20 – 20 Subtraction property of equality
10x = 50 Subtract
\(\frac{10x}{10}\) = \(\frac{50}{10}\) Division property of equality
x = 5 Simplify
Hence, the solution of the equation is x = 5.
Example 2: Write the sentence as an equation and solve for x:
Seven more than four times a number is 35.
Solution:
Seven more than four times a number is 35.
7 + 4x = 35 Write the equation
7 + 4x – 7 = 35 – 7 Subtraction property of equality
4 x = 28 Subtract
\(\frac{4x}{4}\) = \(\frac{28}{4}\) Division property of equality
x = 7 Simplify
Hence, the solution of the equation is x = 7.
Example 3: Find the value of x from the following figure.
Solution:
The angle sum property of a triangle states that the sum of all the interior angles is 180°.
So, x + 2x + ( x + 60° ) = 180° Write the equation
x + 2x + x + 60° = 180° Remove parentheses
4x + 60° = 180° Combine like terms
4x + 60° -60° = 180° -60° Subtraction property of equality
4x = 120° Subtract
\(\frac{4x}{4}\) = \(\frac{120^\circ}{4}\) Division property of equality
x = 30° Simplify
So, the value of x in the figure is 30°.
Example 4: The cost C (in dollars) of manufacturing x hot glue guns are given by the following equation.
C = 25x + 50
If the total cost is $800 dollars, find the total number of hot glue guns that were made.
Solution:
C = 25x + 50 = 800 Write the equation
25x + 50 – 50 = 800 – 50 Subtraction property of equality
25x=750 Subtract
\(\frac{25x}{25}\) = \(\frac{750}{20}\) Division property of equality
x = 30 Simplify
Therefore, 30 hot glue guns can be manufactured for $800.
Example 5: Joel bought a football jersey of a national league team for a certain amount of money. He spent $100 less than twice the money spent on the football jersey to buy a pair of cleats. If the amount spent on the kit is $320, find the cost of the jersey and the cleats.
Solution:
Total amount spent on football kit = $320
Let the amount spent on the jersey be x.
So, the amount spent on the cleats is 2x – 100.
We get the equation as x + 2x – 100 = 320.
x + 2x – 100 = 320 Write the equation
3x-100=320 Group like terms
3x – 100 + 100 = 320 + 100 Subtraction property of equality
3x = 420 Subtract
\(\frac{3x}{3}\) = \(\frac{420}{3}\) Division property of equality
x = 140
So, the cost of the football jersey is $140.
Cost of the pair of cleats = 2x – 100
= 2 x 140 – 100 Substitute the value of x
=280 – 100 = 180
So, Joel spent $140 on the football jersey and $180 on the pair of cleats.
A multi-step equation can be solved by expanding the terms in the parenthesis, combining and collecting the like terms, and then isolating the variables to get the final solution.
To apply the division property of equality, we divide both sides of the equation with the same quantity. The new equation will have the same solution as the original equation. That means, the two equations are equivalent equations.
An expression is a mathematical phrase that contains numbers and variables. Expressions will not have an “equal to” sign. An equation is a mathematical statement that connects two equal expressions with an “equal to” sign.