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Chapter 2: Problem 36

Find the intersection in the \(x y\) -plane of the lines \(y=-4 x+5\) and \(y=5 x-2\).

Short Answer

Expert verified
The intersection point of the two lines in the \(xy\)-plane is \(\left(\frac{7}{9}, \frac{17}{9}\right)\).

Step by step solution

01

Set the equations equal to each other

To find the intersection point, we need to set the y values equal to each other, which means: \[ -4x + 5 = 5x - 2 \]
02

Solve for x

Now we'll solve for x by adding 4x to both sides of the equation and adding 2 to both sides: \[ 5x + 4x = 5 + 2 \] \[ 9x = 7 \] To find the value of x, divide both sides by 9: \[ x = \frac{7}{9} \]
03

Find the y-coordinate

Now that we have the x-coordinate, we can find the y-coordinate by plugging the value of x back into one of our original equations. We can use the first equation: \[ y = -4x + 5 \] \[ y = -4\left(\frac{7}{9}\right) + 5 \] \[ y = -\frac{28}{9} + \frac{45}{9} \] \[ y = \frac{17}{9} \]
04

Write the intersection point

Now that we have both the x and y coordinates, we can write the intersection point as a coordinate pair: \[\left(\frac{7}{9}, \frac{17}{9}\right)\] Thus, the intersection point of the two lines in the \(xy\)-plane is \(\left(\frac{7}{9}, \frac{17}{9}\right)\).

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