## System-of-Equations Word Problems | Purplemath

Systems of Equations Word Problems Date_____ Period____ 1) The school that Lisa goes to is selling tickets to the annual talent show. On the first day of ticket sales the school sold 4 senior citizen tickets and 5 student tickets for a total of \$ The school took in \$ Word Problems Involving Systems of Linear 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. L Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Systems of Equations Word Problems Date_____ Period____ 1) Find the value of two numbers if their sum is 12 and their difference is 4. 4 and 8 2) The difference of two numbers is 3. Their sum is Find the numbers. 5 and 8.

## Systems of Linear Equations and Word Problems – She Loves Math

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 solving word problems with systems of equations 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:, solving word problems with systems of equations.

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, solving word problems with systems of equations, 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 solving word problems with systems of equations 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, solving word problems with systems of equations, 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. There is no general rule for telling which of these things to do: You have to think about what the problem is telling you.

Solve the equations by multiplying the first equation by 25 and subtracting it from the second:. Thenso. How many of each kind of ticket were sold? An investor buys a total of shares of two stocks. How many shares of each stock did the investor buy? SinceI have. The next problem is more complicated than the others, since it involves solving a system of three equations with three variables.

You'll see that I do it by substitution. If you take more advanced courses such as linear algebrayou'll learn methods for solving systems like these which are like the whole equation method. They involve representing the equations using matrices. Phoebe has some cent stamps, some cent stamps, and some 3-cent stamps.

The number of cent stamps is 10 less solving word problems with systems of equations the number of cent stamps, while the number of 3-cent stamps is 5 less than the number solving word problems with systems of equations cent stamps.

How many of each stamp does she have? I'll let x be the number of cent stamps, let y be the number of cent stamps, and let z be the number of 3-cent stamps.

Here's the table, solving word problems with systems of equations. I want to get everything in terms of one variable, so I have to pick a variable to use. Since the last two equations both involve y, I'll do everything in terms of y. I'll solve for x in terms of y:, solving word problems with systems of equations. Plug and into and solve for y:.

Phoebe has 20 cent stamps, 10 cent stamps, and 5 3-cent stamps. The next problem is about numbers. The setup will give two equations, but I don't need to solve them using the whole equation approach as I did in other problems. Since one variable is already solved for in the second equation, I can just substitute for it in the first equation.

The sum of two numbers is The larger number is 14 more than 3 times the smaller number. Find the numbers. Plug into the first equation and solve:. The numbers are 19 and The next set of examples involve simple interest. Here's how it works. At the end of one interest period, the interest you earn is. You now have dollars in your account. Notice that you multiply the amount invested the principal by the interest rate in percent to get the amount of interest earned.

By the way How does "percent" fit the pattern of the earlier problems, where I had things like "dollars per ticket" or "cents per nickel"? In fact, "percent" is short for "per centum", and centum is the Latin word for a hundred.

So "4 percent" means "4 per ". Since "per" translates to division, I getas you probably know from earlier math courses. How much was invested in each account? Bonzo invests some money at interest. How much was invested at each rate? There are various kinds of mixture problems. The first few involve mixtures of different things which cost different amounts per pound. How many pounds of each kind of candy did he use in the mix? How many pounds of raisins and how many pounds of nuts should she use?

Solve the equations by multiplying the first equation by and subtracting it from the second:. Hence, and. She needs 8 pounds of raisins and 9 pounds of nuts. Mixture problems do not always wind up with two equations to solve. Here's an example where the setup gives a single equation. The last line says. Solve for x:. An alloy is a mixture of different kinds of metals, solving word problems with systems of equations.

Suppose you have 50 pounds of an alloy which is silver. Then the number of pounds of pure silver in the 50 pounds is.

That is, the 50 pounds of alloy consists of 10 pounds of pure silver and pounds of other metals. Notice that you multiply the number of pounds solving word problems with systems of equations alloy by the percentage of silver to get the number of pounds of pure silver.

Phoebe mixes an alloy solving word problems with systems of equations silver with an alloy containing silver to make pounds of an alloy with silver. How many pounds of each kind of alloy did she use?

### Solving Systems of Equations Word Problems

Solving Systems of Equations Real World Problems. Wow! 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. Word Problems Involving Systems of Linear 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. Examples of setting up word (or application) problems solved by a system of equations. Show Step-by-step Solutions Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.