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Wize High School Grade 9 Math Textbook > Analytic Geometry: Equation of a Line

Different Forms of the Equation of a Line

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Different Forms of the Equation of a Line

There are 3 different ways we can represent the equation of a line.

1. Slope y-intercept form


2. Standard form


3. Point-slope form


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Example: Converting Between Different Forms of the Equation of a Line

a) Rewrite 2x−3y=62x-3y=6 into slope y-intercept form y=mx+by=mx+b.

2x−3y=6−2x         −2x−3y=−2x+6−3y−3=−2x−3+6−3y=23x−2\begin{array}{rcl} 2x-3y&=&6\\ \colorTwo{-2x~~~~~~~~~}&&\colorTwo{-2x}\\\\ -3y&=&-2x+6\\\\ \dfrac{-3y}{\colorTwo{-3}}&=&\dfrac{-2x}{\colorTwo{-3}}+\dfrac{6}{\colorTwo{-3}}\\\\ y&=&\dfrac{2}{3}x-2 \end{array}

So, the slope y-intercept form is y=23x−2\boxed{y=\dfrac{2}{3}x-2}.


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b) Express y=−54x+72y=-\dfrac{5}{4}x+\dfrac{7}{2} standard form Ax+By+C=0Ax+By+C=0.

We need to first get rid of fractions by multiplying all terms by the common denominator 4:

y×4=−54x×4+72×44y=−5x+14+5x     +5x5x+4y      =14−14−145x+4y−14=0\begin{array}{rcl} y\colorTwo{\times4}&=&-\dfrac{5}{4}x\colorTwo{\times 4}+\dfrac{7}{2}\colorTwo{\times4}\\\\ 4y&=&-5x+14\\ \colorTwo{+5x~~~~~}&&\colorTwo{+5x}\\\\ 5x+4y~~~~~~&=&14\\ \colorTwo{-14}&&\colorTwo{-14}\\\\ 5x+4y-14&=&0 \end{array}

So, the standard form is 5x+4y−14=0\boxed{5x+4y-14=0}.

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c) Rewrite y−1=−2(x+4)y-1=-2(x+4) into slope y-intercept form y=mx+by=mx+b.

y−1=−2(x+4)y−1=−2x−8+1         +1y=−2x−7\begin{array}{rcl} y-1&=&-2(x+4)\\\\ y-1&=&-2x-8\\ \colorTwo{+1}&&\colorTwo{~~~~~~~~~+1}\\\\ y&=&-2x-7 \end{array}

So, the slope y-intercept form is y=−2x−7\boxed{y=-2x-7}

Practice: Different Forms of the Equation of a Line

Given the line 5x−2y=35x-2y=3,

a) Express this line in slope y-intercept form y=mx+by=mx+b.

b) Identify the slope and y-intercept of the line.

Practice: Different Forms of the Equation of a Line

Without graphing or rearranging the equation, determine whether each of the following lines will have a positive slope (the line goes up and to the right) or a negative slope (the line goes down and to the right).

a) −4y=2x+5-4y=2x+5

b) 5x−y=75x-y=7

c) 3x+2y=53x+2y=5

d) y−4=−3(x+1)y-4=-3(x+1)


Practice: Converting Between Linear Equations

Answer the questions that follow each of the following linear equations.
y=−5x+3y=-5x+3
a) Rewrite this equation in standard form.
b) Identify the slope of the line.
c) Identify the y-intercept of the line.
d) True or False. The point (1,−2)(1,-2) is on this line.
e) True or False. The point (0,0)(0, 0) is on this line.

Practice: Different Forms of the Equation of a Line

Wize academy is hosting a school play for its parents and students, and is donating all ticket sales to charity. Tickets for parents cost $7 each and tickets for students cost $5 each. The ticket sales totaled to $2,900.

a) Write an equation in standard form for the total ticket sales. (Use xx to represent the number of parent tickets sold and use yy to represent the number of student tickets sold.)

b) Rewrite the equation in part a) in slope y-intercept form y=mx+by=mx+b.