Equations of Lines

1. The slope of a line through the points $(x_{1},y_{1})$ and $x_{2},y_{2})$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$.

2. Point-Slope Form

The line through the point $(x_1,y_1)$ with slope m has the equation $y-y_1=m(x-x_1)$.

3. Slope-Intercept Form

The line with slope m and y-intercept b has equation $y=mx+b$.

4. Two lines with slopes $m_1$ and $m_2$ are

parallel iff $m_1=m_2$
perpendicular iff $m_1 m_2=-1$.

Ex 1 Find an equation of the line which passes through the point $(4,-7)$ and has slope 3.

Sol Using the Point-Slope Form, we obtain the equation $y-(-7)=3 (x-4)$ or, simplifying, $y+7=3 (x-4)$ or $y=3x-19$.

If instead we use the Slope-Intercept Form, we get the equation $y=3x+b$. To determine b, we substitute $x=4$ and $y=-7$ to obtain $-7=3(4)+b$, so that $b=-19$.

Ex 2 Find an equation of the line which passes through the point $(-2,-6)$ and is parallel to the line $4x-3y=24$.

Sol To find the slope of the line $4x-3y=24$, solve for y to get the equation $y=4/3x-8$; so this line has slope $m=4/3$. Since the line we are seeking is parallel to this line, its slope is also $4/3$, so its equation is $y-(-6)=4/3(x-(-2))$ or $y+6=4/3(x+2)$ or $y=4/3x-10/3$.

Pr 1 Find an equation of the line which passes through the point $(8,-2)$ and has y-intercept 5.

Pr 2 Find an equation of the line which passes through the points $(1,4)$ and $(-3,2)$.

Pr 3 Find an equation of the line which passes through the points $(5,7)$ and $(5,-8)$.

Pr 4 Find an equation of the line which passes through the point $(4,-3)$ and is perpendicular to the line $2x-5y=20$.

Pr 5 Find an equation of the perpendicular bisector of the line segment between the points $(6,-3)$ and $(2,5)$.

Pr 6 Find the slope-intercept form of the equation of the line which passes through the point$(7,-4)$ and which is parallel to the line through the points $(6,3)$ and $(4,-1)$.

Pr 7 Find an equation of the tangent line to the circle $x^2+y^2=25$ at the point $(4,-3)$.

Pr 8 Find the point of intersection of the lines with equations $x-2y=5$ and $2x-5y=11$.

Go to Solutions.

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