# Solving Linear Equations | Best Algebra II Review

This video gives a detailed example of solving a word problem using a linear equation. A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable.

## Linear Equations

Mariel has ten coins in her pocket. All of them are quarters and dimes, and their total value is one dollar and 60 cents. How many of each type of coin does she have? We can solve this by using a system of equations. We’re told that she has ten coins in our pocket and that they’re all quarters and dimes.

If we use D for the number of dimes, and Q for the number of quarters, then one of our equations could be her number of dimes, plus her number of quarters is 10 total coins. Next we are told that the total value is a dollar sixty. Dimes are worth ten cents.

10 times however many dimes she has, plus quarters are worth 25 cents so 25 times however many quarters she has, will give us a total of 160. We’ve eliminated the decimal so we just moved the decimal over two times to the right for all of our numbers. Instead of ten cents we have ten, instead of 25 cents, we have 25, and instead of a 1.60, we have 160.

You could leave the decimals in there if you wanted to. To solve this system we’ve got three options. We could graph both of these and see where they intersect. We could use substitution, or elimination. It’s your choice, for this particular problem I’m going to use substitution. I’m going to solve for D in my first equation, so I can substitute in my second equation.

D is equal to, and I have to subtract Q from both sides, so -Q plus 10. To solve for D, subtract Q from both sides, and you get -Q plus 10. In my second equation, I’m going to use negative Q plus 10 to substitute for D. We have 10 times the quantity, instead of D, -Q plus 10, plus 25Q, is equal to 160. Now we can solve for Q.

To do that first, we need to distribute. 10 times -Q is -10Q, 10 times 10 is 100, plus 100, plus 25Q, equals 160. To solve for Q, we need to combine like terms. -10Q plus 25Q is 15Q, plus 100 is equal to 160. Subtract 100 from both sides. We get that 15Q is equal to 60.

Now we divide both sides by 15. 15 divided by 15 is 1, and 1 times Q is just Q, we get that Q is 4. Wich means she has 4 quarters. How many times does she have? We know that her number of dimes is equal to -Q, so -4, plus 10. Her number of dimes is -4 plus 10, 6. We can say, “Merial has 4 quarters and six dimes in her pocket.”

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Last updated: 09/28/2018