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# Linear Equation Word Problems

A linear equation is an algebraic equation that represents a straight line when graphed on a Cartesian coordinate system.

It typically takes the form:

ax + by = c

where a, b, c are constants and x, y are variables.

The variables x and y each are raised to the power 1 and are not multiplied or divided by each other.

Linear equations can also be represented in slope-intercept form:

y = mx + b where m is the slope and b is the y-intercept.

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Linear equation word problems involve real-world situations where relationships between variables can be described by linear equations. These problems require you to translate the given information into mathematical expressions and equations, solve them to find the unknowns, and then interpret the solutions in the context of the problem.

## Example Questions

1. A car rental company charges a flat fee of \$50 plus \$20 per day for renting a car. Write a linear equation to represent the total cost (C) of renting a car for a certain number of days (d), and find the total cost for renting the car for 5 days.

Solution

Defining Variables:

Let:

C represents the total cost of renting the car.

d represents the number of days the car is rented.

Setting up the Equation:

We know that the total cost (C) consists of two parts: a flat fee of \$50 and an additional charge of \$20 per day. This can be represented by the equation:

C = 50 + 20d

This equation is linear because the relationship between the total cost and the number of days is a straight line.

Solving the Problem:

To find the total cost for renting the car for 5 days, we substitute d = 5 into the equation:

C = 50 + 2(5) = 50 + 100 = 150

So, the total cost for renting the car for 5 days is \$150.

2. A fruit vendor sells apples for \$2 each and oranges for \$3 each. Write a linear equation to represent this and find the total cost of both 5 apples and 3 oranges.

Solution

Cost of each apple = c1 = \$2

Cost of each orange = c2 = \$3

Total cost of 5 apples and 3 oranges is given by the equation 5c1 + 3c2

= 5(2) + 3(3) = 10 + 9 = \$19

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3. Jane’s age is 5 years more than double the age of her brother. The sum of their ages is 35. How old are Jane and her brother?

Solution

Let j represent Jane’s age and b represent her brother’s age

According to the problem,

j = 2b + 5 ….(1) and

j + b = 35….(2)

Plugging j from (1) in (2)

(2b + 5) + b = 35

=> 3b = 30

=> b = 10

=> j = 35 – b = 35 – 10 = 25

4. Two third of a number is 8. Find the number.

Solution

Converting the statement into an equation, we have

2x/3 = 8

=> 2x = 24

x = 12

## Linear Equation Word Problems FAQS

##### What are linear equation word problems?

Linear equation word problems involve real-world scenarios where relationships between variables can be described by linear equations. They require translating verbal descriptions into mathematical expressions and equations.

##### How do you identify linear equations in word problems?

Linear equations typically involve relationships between variables that can be represented by a straight line on a graph. In word problems, look for situations where the relationship between variables is proportional and can be expressed as ax + by = c or y = mx + b.

##### What are some common types of linear equation word problems?

Common types of linear equation word problems include problems involving distance, rate, and time, problems involving age differences, problems involving pricing and revenue, problems involving mixtures, and problems involving geometric figures.

##### How do you solve linear equation word problems?

To solve linear equation word problems, read the problem carefully, identify the variables and their relationships, set up the appropriate linear equation(s), solve the equation(s) algebraically, and interpret the solution in the context of the problem.

##### Can linear equation word problems involve more than two variables?

Yes, linear equation word problems can involve more than two variables. These problems may require setting up and solving systems of linear equations involving multiple variables.

##### What are some strategies for solving linear equation word problems?

Some strategies for solving linear equation word problems include defining variables, drawing diagrams or graphs, breaking down complex problems into smaller steps, and checking solutions for reasonableness.

##### How do you know if your solution to a linear equation word problem is correct?

You can check if your solution to a linear equation word problem is correct by verifying that the solution satisfies all conditions stated in the problem and by substituting the solution back into the original equation to see if it balances.

##### What are the real-life applications of linear equation word problems?

Linear equation word problems have various real-life applications, including problems related to budgeting and finance, problems related to business and economics, problems related to motion and distance, problems related to engineering and construction, and problems related to population dynamics.

##### How can linear equation word problems help develop problem-solving skills?

Linear equation word problems require critical thinking, logical reasoning, and mathematical skills to analyze and solve real-world scenarios. By practicing these problems, students can develop problem-solving strategies and enhance their mathematical fluency.

##### Where can I find resources to practice linear equation word problems?

Resources for practicing linear equation word problems include textbooks, online math websites, educational apps, and math problem-solving books, which offer a variety of problems with solutions for students to practice and improve their skills.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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