Multi Step Equations
Multi step equations are equations that need more than two steps to solve for the variable. We use the same operation on both sides of the equation to such equations. Solving multiple step equations is sometimes complicated when compared to one step or two-step equations as they require multiple steps.
Let us see how to solve multi-step equations and the properties that we use for the same.
|1.||What are Multi Step Equations?|
|2.||Inverse Operations For Solving Multi Step Equations|
|3.||Solving Multi Step Equations|
|4.||Multi Step Equations with Fractions|
|5.||FAQs on Multi Step Equations|
What are Multi Step Equations?
Multi step equations are equations that require more than one operation (applied on both sides) to solve for the required variable. Often word problems also may lead to multi step equations. They look complicated . Here are some examples of multi step equations:
- 2 (4x - 5) = 3x - 7
- –3 [(1/5)t + 1/3] = 9
- 16 - (2x + 1) = (5x - 2) + 13
We already know how to solve one-step and two-step equations. We just extend the same process to solve the multi step equations as well. i.e., we use the properties of equations (like adding, subtracting, multiplying, or dividing both sides by some number/variable such that the equation is balanced) to solve the multi step equations. Let us see how to solve them.
Inverse Operations For Solving Multi-Step Equations
We solve the multi step equations by applying inverse operation on both sides to isolate the variable (making the variable alone on one side of the equation). Note that equality should not be disturbed when we apply any operation. For this, we should apply the same operation on both sides. For example, to solve x + 2 = 3 (of course, this is not a multi step equation), we should subtract 2 from both sides, then we get x + 2 - 2 = 3 - 2 which gives x = 1. Here since we are subtracting 2 from the left side, we should subtract the same number 2 from the right side as well. Wait! Why did we "subtract" 2? Because in the original equation x + 2 = 3, 2 was getting added to x, and to solve for x, we do NOT need + 2 on the left side, so we have subtracted it (as subtraction is the inverse operation of addition). Let us quickly revise the inverse operations:
- Addition and subtraction are inverse operations of each other
- Multiplication and division are inverse operations of each other
- Exponents and roots are inverse operations of each other
Example: square and square root are inverse operations of each other,
cube and cube root are inverse operations of each other, etc).
Applying the same operation on both sides without affecting the equality is proposed by the properties of equations. Here are some examples to understand them.
|x + 1 = 3||Subtract 1 from both sides||x = 2|
|x - 1 = 3||Add 1 on both sides||x = 4|
|2x = 6||Dividing both sides by 2||x = 3|
|x/2 = 6||Multiplying both sides by 2||x = 12|
|x2 = 4||Taking square root on both sides||x = 2|
Solving Multi Step Equations
To solve multi step equations we may need to apply multiple types of inverse operations one after the other (that are mentioned in the previous section). The order of applying inverse operations is very important while solving multi step equations. For example, to solve the equation 2x + 4 = 6, the first step is NOT dividing both sides by 2, rather we subtract 4 from both sides. i.e.,
2x + 4 = 6
Subtracting 4 from both sides,
2x = 2
Dividing both sides by 2,
x = 1
Our ultimate aim is to get just the variable on one side of the equation. We should aim at getting the answer something like "variable = something". Here are the important steps to solve multi step equations:
- Apply distributive property when you have a parenthesis.
- Combine like terms (if any).
- Collect like terms to one side of the equation. i.e., collect variable terms on the left side and the constants on the right side (or vice versa).
- Isolate the variable by inverse operations.
Here is an example to understand these steps.
Example: -2 (x - 3) - 7 = 7x + 11
This is a multi step equation with variable on both sides.
Applying distributive property (i.e., distributing -2 to the terms inside the brackets),
-2x + 6 - 7 = 7x + 11
Combing like terms (i.e., 6 - 7 = -1),
-2x - 1 = 7x + 11
Now our aim is to collect all x terms on the left and all constant on the right.
Subtracting 7x from both sides,
-9x - 1 = 11
Adding 1 on both sides,
-9x = 12
Our aim is fulfilled now. Now, let us divide both side by -9,
x = 12/-9
x = -4/3.
Since x is isolated, it means that we have solved the equation.
Multi Step Equations with Fractions
Sometimes, multi step equations may involve one or more fractions in them. The easiest way of solving such equations is
- Find the LCD (Least Common Denominator) of all the denominators (of both left and right sides).
- Multiply every term on both sides of the equation by LCD.
- Apply inverse operations and isolate the variable.
Here is an example.
Example: Solve (1/4)t + 1/5 = (1/2)t + 5/3
The denominators are 4, 5, 2, and 3. Their LCD is 60. So multiply each term on both sides by 60.
(1/4)t × 60 + 1/5 × 60 = (1/2)t × 60 + 5/3 × 60
15t + 12 = 30t + 100
Now, the equation is free from fractions. We will proceed now. Subtracting 30t and 12 from both sides,
-15t = 88
Dividing both sides by -15,
t = -88/15.
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Multi Step Equations Examples
Example 1: Solve the equation for "s": 3 (4s + 1) = 9.
We can solve the given multi step equation in two ways.
3 (4s + 1) = 9
Dividing both sides by 3,
4s + 1 = 3
Subtracting 1 from both sides,
4s = 2
Dividing both sides by 4,
s = 1/2
3 (4s + 1) = 9
12s + 3 = 9
Subtracting 3 from both sides,
12s = 6
Dividing both sides by 12,
s = 1/2
Answer: The solution is s = 1/2.
Example 2: Solve (1/3) x + 5 = 6x for x.
The given equation is:
(1/3) x + 5 = 6x
Multiply each term on both sides by 3 to avoid the fraction.
x + 15 = 18x
Let us collect the variables on one side.
Subtracting x from both sides,
15 = 17x
Dividing both sides by 17,
x = 15/17
Answer: The solution of the given multi-step equation is x = 15/17.
Example 3: John is 5 years elder than his brother Michael. After 10 years, the sum of their ages is 35. Then how old is Michael now?
This is a multi-step equations word problem.
Let Michael is x years old. Then John's age = x + 5. After 10 years, the sum of their ages is 35. i.e.,
(x + 10) + (x + 5 + 10) = 35
2x + 25 = 35
Subtracting 25 from both sides,
2x = 10
Dividing both sides by 2,
x = 5
Answer: Michael is 5 years old.
FAQs on Multi Step Equations
How Do You Solve Multi Step Equations?
To solve multi step equations, aim at getting the required variable on the left side of the equation. All the other numbers/variables should be removed from the left side by using inverse operations. The inverse operation of addition is subtraction (and vice versa) and the inverse operation of multiplication is division (and vice versa).
How to Solve Multistep Equations With Variables on Both Sides?
If a multi step equation has variables on both sides, then apply inverse operations to get the variable terms on one side and constant terms on the other side. Then solve for x. For example: 3x + 5 = 7x + 6 ⇒ 3x + 5 - 7x = 6 ⇒ -4x + 5 = 6 ⇒ -4x = 6 - 5 ⇒ -4x = 1 ⇒ x = -1/4.
How to Solve Multi Step Equations With Fractions?
If a multi step equation has fractions, then we can eliminate all fractions first by multiplying each term on both sides by LCD of all denominators. Then we can just apply the inverse operations and solve for the variable.
What are the 4 Steps of Multi Step Equations?
To solve multi step equations:
- Expand brackets by using distributive property if any.
- Combine like terms if any.
- Collect like terms of one type on either side.
- Apply inverse operations to isolate the variable.
Where to Find Multistep Equations Worksheets?
We can find multi step equations worksheets by clicking here. You can get varieties of problems (both equations and word problems) by clicking on the given link.
How to Solve Multi Step Equations Word Problems?
To solve multi step equations word problems:
- First, identify the variables with respect to the given context.
- Frame the multi step equation by reading the problem carefully.
- Solve it using the method that is explained on this page.
The multi step equation can also be solved directly through graph and it is often used in linear programming.