Equations Read and Respond

Equations and expressions are both examples of algebra, the branch of math where letters stand in for numbers. These letters are called variables. An algebraic expression is a mathematical phrase that combines numbers, variables and operations. Here's an example: 4b + 20. To turn that expression into an equation, add an equal sign (=) and a value on the other side: 4b + 20 = 100.

Which of these is an equation?

• 9x - 7

• 3(b + 7)

• 4x + 9 = 45

• 2/3 x - 3 > c

The goal of an algebraic equation is to determine the value of the variable. You can do that by isolating it or by getting the variable alone on one side of the equal sign. To solve an equation, use inverse operations. That means undoing or "canceling out" the effects of an operation by using its opposite operation. If you see subtraction on the variable side of the equal sign, use addition, and vice versa. If the coefficient of the variable is multiplication in nature, like 3y, use division (divide by 3) to isolate the variable.

How can you isolate this equation?

x + 6 = 13

• Add 6 to 13

• Add 6 to both sides

• Subtract x from both sides

• Subtract 6 from both sides.

An equation like x + 6 = 13 requires only one step to isolate the variable and find its value: subtract 6 from both sides to get x = 7. Some more complicated equations call for a second step before they can be solved. Here's an example: 18b - 5 = 157. In this equation, there are two items to cancel out: the coefficient (times 18) and the minus 5. If you're familiar with the order of operations, you may want to deal with the times 18 first because multiplication normally comes before subtraction. But when using inverse operations to solve an equation, the order of operations is used backwards: Start with addition and subtraction, then move on to multiplication and division. In this equation, that means you should first add 5 to both sides to cancel out the minus 5. You get 18b = 162. Then divide both sides to cancel out the times 18. The solution: b = 9

What is the first step in isolating the variable in this equation?

5p + 75 = 100

• Subtract 5 from both sides

• Subtract 75 from both sides

• Divide both sides by 75

• Divide both sides by 5

An equation can be used to represent a real-world situation where one number is unknown, not provided or may change. That number is replaced by a variable. Let's say Natalie and her sister Maria were selling chocolates for a school fundraiser. Maria sold 51 chocolates, and they sold 72 chocolates in all. To find out how many chocolates Natalie sold, we can make an equation with a variable: 51 (the number of chocolates Maria sold) + x (the unknown number of chocolates Natalie sold) = 72 (the total number of chocolates sold). Again, that's 51 + x = 72Any variable, from a to z, can be used to represent Natalie's chocolates here; it doesn't have to be x.

Which of situations could model the equation 35 + z = 85?

• Benny earned $85 at a trade show and another $35 selling items at the flea market. How much did he earn in all?

• A diver was 120 feet below sea level and rose up 85 feet to observe some incoming sea life. What was the diver's new elevation?

• Zion scored 35 points on the essay portion of a 2 part test. His overall grade was an 85. How many points did he score on the short response portion of the test?

• Cassie normally has an 85 minute window to arrive at work or she will be late. If her boss adds another 35 minutes to the window, how long of a time frame will she have to arrive to work?

When you've finished solving an equation, it's easy to check your work. Simply plug your answer for the variable into the original equation. Take, for example, the equation 4x + 9 = 45. After isolating the variable, you've come up with x = 9. To check your work, replace the x with 9: 4(9) + 9 = 45. Following the order of operations, multiply 4 by 9 first to get 36 + 9 = 45. Adding 36 plus 9 does equal 45, so you know you've found the right value for x.

Which of these equations is correct when x = -2?

• x + 8 = 5

• 3x - 1 = -10

• 2x - 7 = -11

• -95x = -190