To analyze Jake's estimation of the solution to the system of equations, we need to graph the equations and compare them to his given solution of (-5, -2.5).
The two equations are:
- \( y - 2x = 0 \) or rewritten, \( y = 2x \)
- \( y = 6x + 10 \)
Step 1: Graphing the Equations
Equation 1: \( y = 2x \)
This is a straight line that passes through the origin (0,0) with a slope of 2. Some points on this line include:
- \( (0, 0) \)
- \( (1, 2) \)
- \( (-1, -2) \)
- \( (5, 10) \)
Equation 2: \( y = 6x + 10 \)
This is another straight line with a y-intercept of 10 and a slope of 6. Some points on this line include:
- \( (0, 10) \)
- \( (1, 16) \)
- \( (-1, 4) \)
- \( (-2, -2) \)
Step 2: Finding the Intersection
To find the solution to the system of equations, we can set the two equations equal to each other since at the point of intersection, both equations will have the same \( y \).
Set \( 2x = 6x + 10 \): \[ 2x - 6x = 10 \] \[ -4x = 10 \] \[ x = -\frac{10}{4} = -2.5 \]
Now substitute \( x = -2.5 \) back into one of the original equations to find \( y \): Using \( y = 2x \): \[ y = 2(-2.5) = -5 \]
So the point of intersection, which is the solution to the system of equations, is \( (-2.5, -5) \).
Analyzing Jake's Estimate
Jake said the solution is \( (-5, -2.5) \).
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Mistake in Coordinates:
- Jake transposed the coordinates; the correct solution is \( (-2.5, -5) \), while he wrote it as \( (-5, -2.5) \).
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Interpretation:
- It's possible Jake might have confused the variables or miscalculated the values.
Conclusion
Jake's solution mistake stems from incorrectly placing the coordinates. The correct estimated solution by solving the system is \( (-2.5, -5) \).
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