# Common Core Standards: Math

### Expressions and Equations 8.EE.C.8

8. Analyze and solve pairs of simultaneous linear equations.

Dealing with linear equations means knowing how to handle two-variable equations. We don't just have x anymore. We've added a y as well. Why? It makes life interesting. Plus, it's sort of necessary in order to further their mathematical education. Oh yeah, that.

But wait. A pair of linear equations means more than one. So your students should know how to deal with two different linear equations, each with two variables. The goal here is to help your students understand how to handle multiple linear equations and what they mean.

Basically, make sure they don't drown in all the equations and the variables.

#### Drills

1. Two lines intersect on the coordinate plane. Which of the following is false?

The two linear equations are the same

Since parallel lines never intersect, we know that (C) is can't be false. The intersection point of two different lines is always a point, so (A) is true as well. This point of intersection has the coordinates (x, y), where the values of x and y make both equations true. The only false statement is (B) because two identical linear equations would result in a graph of the same exact line.

2. Two linear equations have a solution. Which of the following is true?

Graphically, this solution takes the form of an intersection point

If two linear equations have a solution, there is a value of x and a value of y that will make both linear equations true. Since these two equations are different, we know (B) is false. We know our answer will take the form of x equaling a number and y equaling a number, so (D) is false too. The graphs of lines represent their possible solutions, and an intersection point would be the solution to both lines, so (C) is right. Just because the two lines intersect doesn't mean they're perpendicular, so (A) isn't necessarily true, either.

3. Two linear equations have no solution. Which of the following is true?

The lines never intersect

If two linear equations have no solution, we can eliminate (D) immediately. No solution means the lines have no points in common, so (A) is right. Two lines that are perpendicular intersect at a 90° angle, so (C) can't be the case here. If the two linear equations were the same, then any x and y that would satisfy one equation would satisfy both, meaning an infinite number of intersection points so (B) is incorrect also.

4. What is the solution to the equations y = 3x and x + y = 16?

x = 4 and y = 12

Since we know that y = 3x, we can substitute 3x for y in the other equation. We'll get x + (3x) = 16 or 4x = 16. If we divide both sides by 4, we get x = 4. Since we know that y = 3x still has to hold true, we can solve for y by plugging in x = 4 and get y = 3(4) = 12. While the other answers all satisfy the equation y = 3x, they don't work for x + y = 16 (since 4, 8, and 12 don't equal 16).

5. What is the solution to the equations x + 2y = 12 and x = 2y?

x = 6 and y = 3

If x = 2y, we can substitute 2y for x in the first equation. That gives us 2y + 2y = 4y = 12. In other words, y = 3. Since x must be twice the value of y, we know that x = 6. While (C) and (D) satisfy the equation x = 2y, they don't work for x + 2y = 12. Answer choice (A) works only for x + 2y = 12, but not for x = 2y. The only answer that satisfies both equations is (B).

6. At what point do the equations x + 5y = 8 and x – 5y = -2 intersect?

(3, 1)

Here, the y's have the same coefficient but opposite signs, so we can add the 2 equations together. Doing so gives us 2x = 6, or x = 3. If we substitute that x = 3 into either equation, our y value should be 1. When plotting points, the coordinates take the form (x, y), so our answer is (A). The points are either switched in (B) and (D) or incorrectly made negative in (C) and (D).

7. At what point do the equations 3x + y = 23 and 2x + y = 16 intersect?

(7, 2)

Since the y coefficients are the same, we can subtract the second equation from the first. This will get us the equation x = 7. Then, substitute that into either equation to get y = 2. While (A) and (B) work for the first equation, (C) is the only one that works for both.

8. Caitlin's favorite café offers two specials: a cappuccino with a donut for \$6 or two cappuccinos and three donuts for \$14. At these prices, how much is a cappuccino and how much is a donut?

Cappuccino: \$4, Donut: \$2

We can use these values and set up a system of equations where c = the price of a cappuccino and d = the price of a donut. Using the two relations, we find that c + d = 6 and 2c + 3d = 14. If we solve these by substitution or by subtraction, we find that the only values that work for both equations are c = 4 and d = 2. In other words, (A) is the right answer.

9. Jamie has decided to get in shape and cut down on eating junk food. Except on weekends. And birthdays. And big important holidays, like Groundhog Day. Anyway, he's trying to decide how to spend his snack Calories today. He can have either three candies and seven pretzels for 100 Calories, or two candies and thirteen pretzels for 100 Calories. Approximately how many more Calories are in a candy than a pretzel?

6

If we let c = the number of Calories in a candy and p = the number of Calories in a pretzel, we can make the equations 3c + 7p = 100 and 2c + 13p = 100. If we solve this system of equations, we end up with c = 24 and p = 4. Comparing these values, we can see that a candy has six times as many Calories as a pretzel because 6p = 6(4) = 24 = c.

10. Olivia wants to bring back the tackiest Mount Rushmore souvenirs she can find for her twelve friends. She can get eight president bobble heads and four Mount Rushmore snow globes for \$88 or six bobble heads and six snow globes for \$84. How much does each souvenir cost?

Bobble head: \$6, Snow globe: \$8