The tables represent two linear functions in a system.
| x | y |
|---|---|
| -4 | 26 |
| -2 | 18 |
| 0 | 10 |
| 2 | 2 |
| x | y |
|---|---|
| -4 | 14 |
| -2 | 8 |
| 0 | 2 |
| 2 | -4 |
What is the solution to this system?
A. (1,0)
B. (1,6)
C. (8,26)
D. (8,-22)
Answer (2)
Find the equation of the first line: y = − 4 x + 10 .
Find the equation of the second line: y = − 3 x + 2 .
Solve the system of equations by setting the two equations equal: − 4 x + 10 = − 3 x + 2 , which gives x = 8 .
Substitute x = 8 into one of the equations to find y : y = − 4 ( 8 ) + 10 = − 22 . The solution is ( 8 , − 22 ) .
Explanation
Problem Analysis We are given two tables representing two linear functions. Our goal is to find the solution to the system of equations formed by these two functions. This means we need to find the point (x, y) where the two lines intersect.
Finding the Equation of the First Line First, we need to find the equations of the two lines. Let's start with the first table. We can use the slope-intercept form of a linear equation, which is y = m x + b , where m is the slope and b is the y-intercept. To find the slope, we can use two points from the table, for example, (-4, 26) and (-2, 18). The slope m is calculated as: m = x 2 − x 1 y 2 − y 1 = − 2 − ( − 4 ) 18 − 26 = 2 − 8 = − 4 So, the slope of the first line is -4. Now we need to find the y-intercept b . We can use one of the points from the table, for example, (0, 10). Plugging this into the equation y = − 4 x + b , we get: 10 = − 4 ( 0 ) + b b = 10 So, the equation of the first line is y = − 4 x + 10 .
Finding the Equation of the Second Line Now let's find the equation of the second line. We'll use the same method. Using the points (-4, 14) and (-2, 8), we can calculate the slope m as: m = x 2 − x 1 y 2 − y 1 = − 2 − ( − 4 ) 8 − 14 = 2 − 6 = − 3 So, the slope of the second line is -3. Now we need to find the y-intercept b . We can use the point (0, 2). Plugging this into the equation y = − 3 x + b , we get: 2 = − 3 ( 0 ) + b b = 2 So, the equation of the second line is y = − 3 x + 2 .
Solving for x Now that we have the equations of both lines, we can solve the system of equations by setting the two equations equal to each other: − 4 x + 10 = − 3 x + 2 To solve for x , we can add 4 x to both sides and subtract 2 from both sides: 10 − 2 = − 3 x + 4 x 8 = x So, x = 8 .
Solving for y Now that we have the value of x , we can substitute it into either equation to find the value of y . Let's use the first equation: y = − 4 x + 10 y = − 4 ( 8 ) + 10 y = − 32 + 10 y = − 22 So, y = − 22 .
Final Answer Therefore, the solution to the system of equations is ( 8 , − 22 ) .
Examples
Systems of linear equations are used in many real-world applications, such as determining the break-even point for a business. For example, suppose a company has fixed costs of $10,000 and variable costs of $5 per unit. The revenue per unit is 15. W ec an se t u p a sys t e m o f e q u a t i o n s t ore p rese n tt h e t o t a l cos t an d t o t a l re v e n u e . L e t x b e t h e n u mb ero f u ni t s . T h e t o t a l cos t i s y = 5x + 10000 an d t h e t o t a l re v e n u e i s y = 15x$. Solving this system of equations will give the break-even point, which is the number of units the company needs to sell to cover its costs. In this case, 5 x + 10000 = 15 x , so 10 x = 10000 , and x = 1000 . The break-even point is 1000 units. This means the company needs to sell 1000 units to cover its costs.
The solution to the system of equations represented by the two tables is the point (8, -22). This was found by determining the equations of the lines from the tables and solving for their intersection. The answer corresponds to option D: (8, -22).
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