To determine the relationship between the two lines represented by the given equations, we can analyze their slopes and intercepts.
- Find the slope of each equation in the slope-intercept form (y = mx + b):
For the first equation: \[ 1.5x + 0.2y = 2.68 \]
Rearranging this gives: \[ 0.2y = -1.5x + 2.68 \] \[ y = -7.5x + 13.4 \]
The slope (m) of the first line is -7.5.
For the second equation: \[ 1.6x + 0.3y = 2.98 \]
Rearranging this gives: \[ 0.3y = -1.6x + 2.98 \] \[ y = -\frac{16}{3}x + \frac{2.98}{0.3} \] Calculating the constant: \[ \frac{2.98}{0.3} \approx 9.9333 \] So, \[ y \approx -5.3333x + 9.9333 \]
The slope (m) of the second line is approximately -5.33.
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Compare the slopes: The slopes are different (-7.5 and -5.33), so the lines are not parallel.
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Check if there is a point of intersection: To find if they intersect, we can try solving the two equations together. We can set them equal to find the \(x\) and \(y\) values.
Solving the system of equations:
- \( 1.5x + 0.2y = 2.68 \) (Multiply by 10) → \( 15x + 2y = 26.8 \)
- \( 1.6x + 0.3y = 2.98 \) (Multiply by 10) → \( 16x + 3y = 29.8 \)
Now we can use the method of elimination or substitution, but instead, we'll use a quick numerical check:
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If we check the proposed intersection points:
- For (1.6, 1.4):
- First equation: \( 1.5(1.6) + 0.2(1.4) = 2.4 + 0.28 = 2.68 \) (True)
- Second equation: \( 1.6(1.6) + 0.3(1.4) = 2.56 + 0.42 = 2.98 \) (True)
- Therefore, they indeed intersect at (1.6, 1.4).
- For (1.6, 1.4):
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Checking the second proposed point (3.1, 0.5):
- First equation: \( 1.5(3.1) + 0.2(0.5) = 4.65 + 0.1 = 4.75\) (False)
- This point does not satisfy the first equation.
So the valid conclusion is:
The lines intersect at (1.6, 1.4).
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