The tables represent two linear functions in a system.
| x | y |
|---|---|
| -6 | -22 |
| -3 | -10 |
| 0 | 2 |
| 3 | 14 |
| x | y |
|---|---|
| -6 | -30 |
| -3 | -21 |
| 0 | -12 |
| 3 | -3 |
What is the solution to this system?
A. [tex]\left(-\frac{13}{3},-25\right)[/tex]
B. [tex]\left(-\frac{14}{3},-54\right)[/tex]
C. (-13, -50)
D. (-14, -54)
Answer (2)
Determine the equations of the two lines from the tables: y = 4 x + 2 and y = 3 x − 12 .
Set the two equations equal to each other: 4 x + 2 = 3 x − 12 .
Solve for x: x = − 14 .
Substitute x into one of the equations to solve for y: y = 4 ( − 14 ) + 2 = − 54 . The solution is ( − 14 , − 54 ) .
Explanation
Understanding the Problem We are given two tables representing two linear functions. Our goal is to find the solution to the system of equations formed by these linear functions. This means we need to find the point (x, y) where the two lines intersect.
Finding the Equations First, we need to determine the equations of the two lines. We can find the slope (m) and y-intercept (b) for each line using the data from the tables.
Equation of the First Line For the first table: We can calculate the slope using two points, for example, (-6, -22) and (-3, -10): m 1 = − 3 − ( − 6 ) − 10 − ( − 22 ) = 3 12 = 4 The y-intercept is the y-value when x = 0, which is given as 2 in the table. So, b 1 = 2 .
Therefore, the equation of the first line is: y = 4 x + 2
Equation of the Second Line For the second table: We can calculate the slope using two points, for example, (-6, -30) and (-3, -21): m 2 = − 3 − ( − 6 ) − 21 − ( − 30 ) = 3 9 = 3 The y-intercept is the y-value when x = 0, which is given as -12 in the table. So, b 2 = − 12 .
Therefore, the equation of the second line is: y = 3 x − 12
Solving the System of Equations Now we have a system of two equations: y = 4 x + 2 y = 3 x − 12 To find the solution, we set the two equations equal to each other: 4 x + 2 = 3 x − 12
Finding the Solution Solving for x: 4 x − 3 x = − 12 − 2 x = − 14 Now, substitute the value of x into either equation to find y. Let's use the first equation: y = 4 ( − 14 ) + 2 y = − 56 + 2 y = − 54
Final Answer Therefore, the solution to the system of equations is (-14, -54).
Examples
Systems of linear equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell a product at a certain price, we can set up a system of equations to find the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve these systems helps in making informed business decisions. Another example is in physics, where systems of equations can be used to analyze forces acting on an object. By setting up equations based on the forces, we can solve for unknown variables and understand the object's motion.
The solution to the system of linear functions represented by the two tables is (-14, -54). This was found by determining the equations of the functions, setting them equal to each other, and solving for the coordinates. Therefore, the correct answer is option D.
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