Forms of Linear Equations

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Point-slope and standard form
7:57 — by Khan Academy

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Key Questions

  • There are 4 forms of linear equations, they are:

  • We can use the x- and y-intercepts to graph linear equations in standard form (#Ax+By=C#).

    For example, let's graph #3x+4y=12#.

    To find the x-intercept we substitute 0 for y in the equation (every point on the x-axis has a y-coordinate of 0).

    #3x+4(0)=12=>3x=12=>x=4#

    So, the x-intercept is 4. Now let's find the y-intercept by substituting 0 for x in the equation (similarly, every point on the x-axis has a y-coordinate of 0).

    #3(0)+4y=12=>4y=12=>y=3#

    So, the y-intercept is 3. We can now plot both points in the coordinate plane and draw a line through them.

    See image below.

    enter image source here

  • The standard form of linear equation is ax+by=c.
    For example:
    y = 5x + 2
    To express this in standard form of linear equation just simply identify what's a, b, and c. And in this expression :
    a= 5 ; b= 1 and c=2. Therefore, the standard form is -5x+y=2 (Note that the value of b become -3 because you need its subtraction property in order to substitute it). However, the value of "a" shouldn't be negative so you need to multiply it by -1 so the answer will be 5x-y=-2.

  • The point slope form of an equation of a line is
    #(y-y_1)=m(x-x_1)# where m is the slope and # (x_1,y_1)# is a point on the line.

    The standard form of the equation of a line is given as #Ax+By=C# where A, B and C are all integers.

    Example: Change #(y-2)=5(x+3)# to standard form.

    First, we distribute 5 to each and every term inside the group at the right side;

    #(y-2)=5(x+3)#
    #y-2=5x+15# Note: Always make A positive in every standard form equation

    We transpose y at the right side instead of 5x at the left, just to make A positive.Also, transpose 15 at the left side.

    #-2-15=5x-y#
    #-17=5x-y#
    Through commutative property of equality, this equation is the same as

    #5x-y=-17# the standard form of the equation.

    Now, how we change point- slope form to slope- intercept form?

    The slope- intercept form of an equation of a line is given as

    #y=mx+b# where m is slope and b is the y- intercept.

    Given #(y-2)=5(x+3)#, change it to slope- intercept form

    Again, distribute 5 in each and every term inside the parenthesis at the right side of the equality.

    #y-2=5x+15#

    Then transpose -2 at the right side, the equation becomes

    #y=5x+15+2#

    #y=5x+17# where m= 5 and b=17.

    Done!

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