# Word Problems using Systems of Equations

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A series of free Basic Algebra Lessons.

In this lesson, we will learn

• how to solve age word problems
• how to solve sum of digits word problems
• how to solve percent concentration word problems
• how to solve selling multiple items word problems

### Age Word Problems

Setting up a system of equations is usually effective when solving age word problems involving the ages of two people. Knowing how to solve systems of equations is necessary to solve these types of problems. Other types of word problems involving systems of equations include mixture problems, money word problems, work word problems and rate word problems.

### Sum of Digits

One difficult type of Algebra problem are word problems asking about the sum of digits in a two digit number. Solving systems of equations by substitution is one of the most common ways to solve sum of digits problems. Other types of word problems using systems of equations include mixture problems, money word problems, work word problems and rate word problems.
Algebra word problem involving digits in a two digit number. Solve an Algebra word problem involving digits in a three digit number. A little more difficult.

### Percent Concentration

To solve word problems involving percent concentration amounts, knowledge of solving systems of equations and percents is necessary. Percent concentration word problems involve solving for amounts and concentrations when mixing two liquids with differing percent concentration amounts. Other types of word problems using systems of equations include mixture problems and money word problems.
Example:
Explains using picture how to solve this percent mixture problem using one variable: How many pounds of dogfood that is 50% rice must be mixed with 400 pounds of dogfood that is 80% rice to make a dogfood that is 75% rice? Explains using picture how to solve this percent mixture problem using one variable:
Example:
How many pounds of dogfood that is 50% rice must be mixed with 400 pounds of dogfood that is 80% rice to make a dogfood that is 75% rice? Explains using picture how to solve this percent mixture problem using one variable:
Example:
Five pounds of candy that is 20% chocolate is combined with a candy that is 40% chocolate. How many pounds of the candy that is 40% chocolate should be used to get a candy that is 25% chocolate? Explains using picture how to solve this percent mixture problem using one variable:
Example:
How much water should be added to 30 cups of juice that is 70% juice to get a diluted mixture that is only 60% juice?

### Selling Multiple Items

To solve word problems where someone is selling multiple items, a background knowledge of setting up a system of equations is necessary. Problems where someone is selling multiple items are common on standardized tests such as the ACT and the SAT. These types of problems are similar to rate word problems, work word problems and money word problems. Example:

Suppose your marching band sponsored a fundraiser. For selling 205 packages of gift wrap and boxes of greeting cards, the band earns \$1,084. The profit for selling gift wrap is \$4 per item and \$10 per box for the greeting cards. How many of each type of package was sold? This problem can certainly be solved using a system of equations. We will do it using elimination.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.