To find the solution of the system of equations given by \( y = 4z - 10 \) and \( y = -2x + 8 \), we first need to find the point where the two lines intersect, which represents the solution to the system.
Since the equations are in terms of \( y \), we can set them equal to each other:
\[ 4z - 10 = -2x + 8 \]
To express \( z \) in terms of \( x \), we rearrange the equation:
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Add 10 to both sides: \[ 4z = -2x + 18 \]
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Divide by 4: \[ z = -\frac{1}{2}x + \frac{9}{2} \]
Now we can express \( y \) in terms of \( x \) as well by substituting different values of \( x \):
To find a specific solution, we can solve for both equations.
From equation \( y = -2x + 8 \), we can input various \( x \) values to find corresponding \( y \) values, or vice versa.
However, to find a specific intersection, we would typically solve for \( x \) and then determine \( y \): For example, if we want to set \( z = 0 \) to find \( x \):
From \( z=0 \): \[ y = -\frac{1}{2}(0) + \frac{9}{2} = \frac{9}{2} = 4.5 \] And from \( y = -2x + 8 \): \[ 4.5 = -2x + 8 \implies -2x = 4.5 - 8 \implies -2x = -3.5 \implies x = 1.75 \]
Now substituting \( x = 1.75 \) into one of the equations for \( z \): \[ z = -\frac{1}{2}(1.75) + \frac{9}{2} \implies z = -0.875 + 4.5 \implies z = 3.625 \]
The solution of the system can be expressed as \( (x, y, z) = (1.75, 4.5, 3.625) \).
Thus, the point of intersection or solution of the system of equations can be summarized in a more standard form as: \[ \text{(x, y) intersection point: } (1.75, 4.5) \] since we have not defined "z" in the original equations explicitly when solving.
So, in answer to your question, the solution of the system of equations is: \((1.75, 4.5)\) if only considering the x and y intersections.