Understanding Linear Equations and Graphs

During the holiday season, Ameera and Christos are planning to decorate their houses with lights. They want the lights to be arranged in parallel lines. Which of the following pairs of equations represent the paths of the lights that will be parallel?

Tags

GA.8.FGR.7.5

During a holiday sale, Jeremiah buys two types of gift baskets. The first type contains 4 chocolates and 1 candle, and the second type contains 2 chocolates and 3 candles. If Jeremiah buys a combination of these baskets and ends up with 9 chocolates and 12 candles, how many of each type of basket did he buy?

Solve:

4x-y=9

2x+3y=12

Answer explanation

To solve the system, substitute y from the first equation into the second. From 4x - y = 9, we get y = 4x - 9. Substituting into 2x + 3(4x - 9) = 12 gives 14x - 27 = 12, leading to x = 2. Then, y = 2. Thus, the solution is (2, 2).

Tags

GA.8.FGR.7.1

Answer explanation

To find the intersection, set the equations equal: 3x - 2 = -x + 6. Solving gives x = 2. Substitute x back into either equation to find y: y = 3(2) - 2 = 4. Thus, the intersection point is (2, 4).

Tags

GA.8.FGR.7.2

During the holiday season, Sasha and Skyler are planning to decorate their house with lights. They want to hang two sets of lights such that they form perpendicular lines on the wall. Which of the following pairs of equations represent the paths of the lights that will be perpendicular?

Answer explanation

The slopes of perpendicular lines are negative reciprocals. The slope of \(y = 3x + 1\) is 3, and the slope of \(y = -\frac{1}{3}x + 4\) is -\frac{1}{3}. Since \(3 \times -\frac{1}{3} = -1\), they are perpendicular.

Tags

GA.8.FGR.7.5

During a holiday sale, Jeremiah and Brianna sold a total of 8 items consisting of hats and scarves. Each hat costs $3 more than a scarf. If the total cost of the hats and scarves is $5, how many of each item did they sell?

Solve:

x+2y=8

3x-y=5

Answer explanation

To solve the system, substitute $y$ from the first equation into the second. From $x + 2y = 8$, we get $y = (8 - x)/2$. Substituting into $3x - y = 5$ gives $3x - (8 - x)/2 = 5$. Solving yields $x = 3$ and $y = 2$, so the solution is (3, 2).

Tags

GA.8.FGR.7.1

Answer explanation

To find the solution, substitute the x-value from (1, 4) into both equations. For y = x + 3, y = 1 + 3 = 4. For y = -2x + 6, y = -2(1) + 6 = 4. Both equations yield y = 4, confirming (1, 4) is the solution.

Tags

GA.8.FGR.7.2

During a holiday parade, Jaylah and Amai are organizing two floats that are moving along different paths. Which of the following pairs of equations represent paths that are neither parallel nor perpendicular to each other?

Answer explanation

The lines represented by y = -x + 4 and y = x + 2 have slopes of -1 and 1, respectively. Since their slopes are neither equal (not parallel) nor negative reciprocals (not perpendicular), they represent neither parallel nor perpendicular lines.

Tags

GA.8.FGR.7.5

During a holiday sale, Sasha and Bryce are shopping for gift baskets. Basket A contains 2 chocolates and 1 candle, and Basket B contains 1 chocolate and 2 candles. If they buy a total of 7 chocolates and 4 candles, how many of each type of basket did they purchase?

Solve:

2x+y=7

x-2y=-4

Answer explanation

To solve the system, substitute y from the first equation into the second. From 2x + y = 7, we get y = 7 - 2x. Substituting into x - 2y = -4 gives x - 2(7 - 2x) = -4. Solving this yields x = 3, y = 1, so the solution is (3, 1).

Tags

GA.8.FGR.7.1

Answer explanation

To find the intersection, set the equations equal: 4x - 1 = -x + 9. Solving gives x = 2. Substituting x back into either equation yields y = 7. Thus, the point of intersection is (2, 7).

Tags

GA.8.FGR.7.2

Answer explanation

To find the solution, graph both equations. The lines intersect at approximately (1, 1), which satisfies both equations. Thus, the correct answer is (1, 1).

Tags

GA.8.FGR.7.3