Which is the approximate solution to the system [tex]y=0.5 x+3.5[/tex] and [tex]y=-\frac{2}{3} x+\frac{1}{3}[/tex] 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 point into the given equations.
Check which point satisfies both equations approximately.
Point (-2.7, 2.1) gives values closest to the actual y-value in both equations.
The approximate solution is ( − 2.7 , 2.1 ) .
Explanation
Analyze 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 from the given options: (-2.7, 2.1), (-2.1, 2.7), (2.1, 2.7), (2.7, 2.1).
Substitute each point into the equations Let's substitute each point into both equations and see which one satisfies both equations approximately.
Point (-2.7, 2.1):
Equation 1: y = 0.5 ( − 2.7 ) + 3.5 = − 1.35 + 3.5 = 2.15
Equation 2: y = − 3 2 ( − 2.7 ) + 3 1 = 1.8 + 0.333... = 2.133...
This point seems close to satisfying both equations.
Point (-2.1, 2.7):
Equation 1: y = 0.5 ( − 2.1 ) + 3.5 = − 1.05 + 3.5 = 2.45
Equation 2: y = − 3 2 ( − 2.1 ) + 3 1 = 1.4 + 0.333... = 1.733...
This point does not satisfy both equations well.
Point (2.1, 2.7):
Equation 1: y = 0.5 ( 2.1 ) + 3.5 = 1.05 + 3.5 = 4.55
Equation 2: y = − 3 2 ( 2.1 ) + 3 1 = − 1.4 + 0.333... = − 1.066...
This point does not satisfy both equations.
Point (2.7, 2.1):
Equation 1: y = 0.5 ( 2.7 ) + 3.5 = 1.35 + 3.5 = 4.85
Equation 2: y = − 3 2 ( 2.7 ) + 3 1 = − 1.8 + 0.333... = − 1.466...
This point does not satisfy both equations.
Find the approximate solution Comparing the results, the point (-2.7, 2.1) gives values closest to the actual y-value in both equations. Therefore, it is the approximate solution.
State the 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 + 100 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. Another example is in physics, where systems of equations can be used to solve for unknown forces or velocities in a system.
The approximate solution to the system of equations is (-2.7, 2.1), as this point satisfies both equations most closely. All other options do not provide results that approximate the values found in the equations. Therefore, the chosen option is A. (-2.7, 2.1).
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