write an equation of a line that passes through the point (7,3) and is parallel to the line y=-2/3x + 3
Answer:
\(y=-\dfrac{2}{3} x+\dfrac{23}{3}\)
Step-by-step explanation:
Recall that parallel lines share the same slope value.
Graphing two lines of the same slope and different y-intercepts produces two lines that never intersects. (Try it out on either a physical or digital graphing calculator!)
Another this can be explained is by imagining a systems of equations. If two lines are parallel, they shouldn't have a solution. If we have say the equations \(y=2x+5\) and \(y=2x+10\), using substitution to solve the system we'd get,
\(2x+10=2x+5\\2x-2x=5-10\\0x=-5\) ,
which there is no possible x value that can make the last equation true, thus no solutions.
So, from this example this confirms that two lines with the same slope regardless of their y-intercepts will have no solutions, implying that they're parallel.
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Applying what we know, if the line in this problem is parallel to the line \(y=-\dfrac{2}{3} x+3\), then our new line must also have a slope of \(-\dfrac{2}{3}\)!
Recalling the general format of a linear equation, \(y=mx+b\), where in our case, y and x stays unchanged, and m is already found, all there's left is to find the y-intercept or b value.
How can we do that?
Knowing that a coordinate pair is given to us that's on our new line or is a solution to our line (plugging in the x-coordinate into the new line's equation produces its y-coordinate), we can plug their x and y values into our tentative equation, rearrange and solve for b!
\(3=-\dfrac{2}{3} (7)+b\)
\(3=-\dfrac{14}{3}+b\)
\(3+\dfrac{14}{3}=b\)
\(\dfrac{9}{3} +\dfrac{14}{3} \\\implies \dfrac{23}{3} =b\\\)
Now the linear equation of our new line is complete!