Forms of Linear Equations

Point-slope and standard form

Tip: This isn't the place to ask a question because the teacher can't reply.

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 ($A x + B y = C$).

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

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).

$3 x + 4 \left(0\right) = 12 \implies 3 x = 12 \implies x = 4$

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).

$3 \left(0\right) + 4 y = 12 \implies 4 y = 12 \implies y = 3$

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.

• 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
$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$ where m is the slope and $\left({x}_{1} , {y}_{1}\right)$ is a point on the line.

The standard form of the equation of a line is given as $A x + B y = C$ where A, B and C are all integers.

Example: Change $\left(y - 2\right) = 5 \left(x + 3\right)$ to standard form.

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

$\left(y - 2\right) = 5 \left(x + 3\right)$
$y - 2 = 5 x + 15$ 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.

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

$5 x - y = - 17$ 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

$y = m x + b$ where m is slope and b is the y- intercept.

Given $\left(y - 2\right) = 5 \left(x + 3\right)$, change it to slope- intercept form

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

$y - 2 = 5 x + 15$

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

$y = 5 x + 15 + 2$

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

Done!

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