Which is the approximate solution to the system $y=0.5 x$ +3.5 and $y=-\frac{2}{3} x+\frac{1}{3}$ shown on the graph?
A. (-2.7, 2.1)
B. (-2.1, 2.7)
C. (2.1, 2.7)
D. (2.7, 2.1)
Answer (2)
Substitute each of the given (x, y) pairs into the two equations.
Check which pair satisfies both equations approximately.
Option (-2.7, 2.1) satisfies both equations: 2.1 ≈ 0.5 ( − 2.7 ) + 3.5 and 2.1 ≈ − 3 2 ( − 2.7 ) + 3 1 .
The approximate solution is ( − 2.7 , 2.1 ) .
Explanation
Understanding the problem We are given a system of two linear equations:
y = 0.5 x + 3.5
y = − 3 2 x + 3 1
We need to find the approximate solution to the system of equations from the given options: (-2.7, 2.1), (-2.1, 2.7), (2.1, 2.7), (2.7, 2.1). The solution is a pair of (x, y) values that satisfy both equations.
Testing the options Let's test each option to see which one satisfies both equations.
Option 1: (-2.7, 2.1)
Equation 1: 2.1 = 0.5 ( − 2.7 ) + 3.5 = − 1.35 + 3.5 = 2.15
Equation 2: 2.1 = − 3 2 ( − 2.7 ) + 3 1 = 1.8 + 0.333... = 2.133...
Both equations are approximately satisfied.
Option 2: (-2.1, 2.7)
Equation 1: 2.7 = 0.5 ( − 2.1 ) + 3.5 = − 1.05 + 3.5 = 2.45
Equation 2: 2.7 = − 3 2 ( − 2.1 ) + 3 1 = 1.4 + 0.333... = 1.733...
Neither equation is satisfied.
Option 3: (2.1, 2.7)
Equation 1: 2.7 = 0.5 ( 2.1 ) + 3.5 = 1.05 + 3.5 = 4.55
Equation 2: 2.7 = − 3 2 ( 2.1 ) + 3 1 = − 1.4 + 0.333... = − 1.066...
Neither equation is satisfied.
Option 4: (2.7, 2.1)
Equation 1: 2.1 = 0.5 ( 2.7 ) + 3.5 = 1.35 + 3.5 = 4.85
Equation 2: 2.1 = − 3 2 ( 2.7 ) + 3 1 = − 1.8 + 0.333... = − 1.466...
Neither equation is satisfied.
Finding the solution From the calculations above, only the first option (-2.7, 2.1) approximately satisfies both equations.
Final Answer The approximate solution to the system of equations is (-2.7, 2.1).
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is y = 2 x + 1000 (where x is the number of units produced) and its revenue function is y = 5 x , solving this system of equations will give the number of units the company needs to sell to break even. Understanding how to solve systems of equations is crucial for making informed business decisions.
The approximate solution to the system of equations is A: (-2.7, 2.1), as it satisfies both equations approximately when substituted. Other options do not satisfy the equations. Therefore, A is the correct choice.
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