### Linear equation word problems substitution

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The following are some examples and solutions for algebra word problems that you will commonly encounter in grade 8. How to write algebra word problems into systems of linear equations and solve systems of linear equations using elimination and substitution methods?

Example: Devon is going to make 3 shelves for her father. He has a piece of lumber 12 feet long. She wants the top shelf to be half a foot shorter than the middle shelf, and the bottom shelf to be half a foot shorter than twice the length of the top shelf. How long will each shelf be if she uses the entire 12 feet of wood? Show Step-by-step Solutions. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics.

Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.Some word problems require the use of systems of linear equations.

Here are clues to know when a word problem requires you to write a system of linear equations:. Such problems often require you to write two different linear equations in two variables. Typically, one equation will relate the number of quantities people or boxes and the other equation will relate the values price of tickets or number of items in the boxes. Use substitutionelimination or graphing method to solve the problem. How much was the admission for each child and adult?

Let x represent the admission cost for each child. Let y represent the admission cost for each adult. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials.

Writing Systems of Linear Equations from Word Problems Some word problems require the use of systems of linear equations. Here are clues to know when a word problem requires you to write a system of linear equations: i There are two different quantities involved: for instance, the number of adults and the number of children, the number of large boxes and the number of small boxes, etc. Here are some steps to follow: 1. Understand the problem. Understand all the words used in stating the problem.

Understand what you are asked to find. Familiarize the problem situation. Translate the problem to an equation. Assign a variable or variables to represent the unknown. Clearly state what the variable represents.

Carry out the plan and solve the problem. Subtract the second equation from the first. Subjects Near Me. Download our free learning tools apps and test prep books.My neighbor has both chickens and roosters. He has a total of 31 birds. The number of chickens is ten more than twice the number of roosters.

## Writing Systems of Linear Equations from Word Problems

How many chickens does he have? The difference between two numbers is twice the smaller number. The smaller number is 10 less than the larger number. Find the smaller number. The sum of two numbers is 19 One number is 1 15 more than the other.

What are the two numbers? The shortest side of a triangle has length 16 inches. The perimeter of the triangle is nineteen less than three times the longest side. The remaining side is eight inches shorter than the longest side. What are the side lengths of the triangle? Bob is three times as old as Bart. Bob is also twenty-two years older than Bart will be in two years. How old is each? In this lesson we focus on how to solve a simple system of equations in which one variable can easily be written in terms of another variable.

This is the simplest kind of system of equations to solve.

### Word Problems Involving Systems of Linear Equations

In fact, many students will solve these kinds of problems without even thinking of it as a system of equations. For this lesson, we require students to use two variables or moremaking these problems a nice transition from single-variable problems to systems problems. Each of these systems is easily solved using substitution. Slide 1 : Discuss with students some of the strategies involved in solving algebra problems.You have learned many different strategies for solving systems of equations!

First we started with Graphing Systems of Equations. Then we moved onto solving systems using the Substitution Method. In our last lesson we used the Linear Combinations or Addition Method to solve systems of equations.

Now we are ready to apply these strategies to solve real world problems! Are you ready? First let's look at some guidelines for solving real world problems and then we'll look at a few examples.

Follow along with me. Having a calculator will make it easier for you to follow along. You are running a concession stand at a basketball game. You are selling hot dogs and sodas. You sold a total of 87 hot dogs and sodas combined. You must report the number of hot dogs sold and the number of sodas sold. How many hot dogs were sold and how many sodas were sold?

In this problem, I don't know how many hot dogs or sodas were sold. So this is what each variable will stand for. Usually the question at the end will give you this information. Write two equations. One equation will be related to the price and one equation will be related to the quantity or number of hot dogs and sodas sold. We can choose any method that we like to solve the system of equations.

I am going to choose the substitution method since I can easily solve the 2nd equation for y.One number is greater than thrice the other number by 2. If four times the smaller number exceeds the greater by 5, find the numbers. Solution :. Let x be the smaller number and y be the greater number. Then, we have. Subtract 3x from each side. Substitute 7 for x in 1.

So, the numbers are 7 and The ratio of income of two persons is 1 : 2 and the ratio of their expenditure is 3 : 7.

Find the cost of each chair. Let x and y be the costs of each chair and table respectively. Subtract 9x from each side. Substitute - 9x for 6y in 2. Subtract from each side.

Divide each side ny A father's age is three times the sum of the age of his two son. After 5 years, his age will be twice the sum of the ages of his two sons. Find the present age of the father. Let x be the present age of the father and y be the sum of the present ages of the two sons. At present :. After 5 years :. If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes 1.

Find the fraction. Add 8 to each side. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. You can also visit our following web pages on different stuff in math. Variables and constants. Writing and evaluating expressions. Solving linear equations using elimination method.

### Word Problems

Solving linear equations using substitution method. Solving linear equations using cross multiplication method. Solving one step equations.Many word problems will give rise to systems of equations that is, a pair of equations like this:. You can solve a system of equations in various ways.

In many of the examples below, I'll use the whole equation approach. To review how this works, in the system above, I could multiply the first equation by 2 to get the y-numbers to match, then add the resulting equations:. If I plug intoI can solve for y:. In some cases, the whole equation method isn't necessary, because you can just do a substitution. You'll see this happen in a few of the examples.

The first few problems will involve items coins, stamps, tickets with different prices. This is common sense, and is probably familiar to you from your experience with coins and buying things. But notice that these examples tell me what the general equation should be: The number of items times the cost or value per item gives the total cost or value. This is where I get the headings on the tables below.

You'll see that the same idea is used to set up the tables for all of these examples: Figure out what you'd do in a particular case, and the equation will say how to do this in general.

If there are twice as many nickels as pennies, how many pennies does Calvin have? How many nickels? In this kind of problem, it's good to do everything in cents to avoid having to work with decimals. So Calvin has cents total. Let p be the number of pennies.

There are twice as many nickels as pennies, so there are nickels. I'll arrange the information in a table. Be sure you understand why the equations in the pennies and nickels rows are the way they are: The number of coins times the value per coin is the total value.

If the words seem too abstract to grasp, try some examples:. If you have 3 nickels, they're worth cents. If you have 4 nickels, they're worth cents. If you have 5 nickels, they're worth cents. So if you have nickels, they're worth cents. The total value of the coins is the value of the pennies plus the value of the nickels.

So I add the first two numbers in the last column, then solve the resulting equation for p:. Therefore, he has nickels. Tables for problems. I'll often arrange the equations for word problems in a tableas I did above. The number of things will go in the first column. This might be the number of tickets, the time it takes to make a trip, the amount of money invested in an account, and so on. The value per item or rate will go in the second column.

This might be the price per ticket, the speed of a plane, the interest rate in percent earned by an investment, and so on. The total value or total amount will go in the third column. This might be the total cost of a number of tickets, the distance travelled by a car or a plane, the total interest earned by an investment, and so on.

There are many correct ways of doing math problems, and you don't have to use tables to do these problems. But they are convenient for organizing information and they give you a pattern to get started with problems of a given kind e.

In some cases, you add the numbers in some of the columns in a table. In other cases, you set two of the numbers in a column equal, or subtract one number from another.Oct 31, Linear Equations Word Problems.

Word problems for systems of linear equations are troublesome for most of the students in understanding the situations and bringing the word problem into equations. We tried to explain the trick of solving word problems for equations with two variables with an example. Find the number of adult tickets and child tickets sold on Saturday. Step We are asked to find the number of adult tickets and children tickets sold. Let x be the number of adult tickets sold and y the number of child tickets sold on Saturday.

Now, we got a system of two linear equations in two variables. Solve the above two linear equations to find the value of x and y. Multiply equation 1 with -2 and add the resulting equation and equation 2 to eliminate the variable y.

Divide by 2 on both the sides and simplify. Let us substitute in equation 1. So, it becomes. Subtract from both sides of the equation and simplify. So, the solution for the given system of equations is, which means adult tickets and child tickets were sold on Saturday. Note: The above problem can be also solved using substitution method since the coefficients of x and y in the first equation is 1. Practice Problems:. The shop keeper strictly told that there will not be any discounts. What is the cost of an apple and an orange?

Ana writes test to upgrade her level. The test has 25 questions for a total score of points. Among the 25 questions, each multiple choice questions carries 3 points and the descriptive type questions carries 8 points.

How many multiple choice questions and descriptive type questions are there in the test? The sum of the digits of a two digit number is 7. When the digits are reversed, then number is decreased by 9. Find the number.

## How Do You Solve a System of Equations Using the Substitution Method?

The perimeter of a rectangle is 12 meters. Find the dimensions of the rectangle. This can also be solved just with one variable. The sum of two numbers is When three times the first number is added to 5 times the second number, the resultant number is Find the two numbers.

Having problem in solving any of these questions? Post your doubt in comment. We will help you how to solve this. Related Articles:. Posted by Mr.