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Functions
Standard Form: At a Glance

Introduction to Standard Form:

A linear equation in standard form is an equation that looks like

ax + by = c

where a, b, and c are real numbers and a and b aren't both zero. c can be zero if it wants. It's the favorite child, so it gets special privileges.

If only a is 0, the equation can be rewritten to look like

y = (some number).

If only b is 0, the equation can be rewritten to look like

x = (some number).

For example, the equation 3y = 8 is equivalent to the equation , which is also in standard form (with b = 1).

Meanwhile, the equation 2x = 4 is equivalent to the equation x = 2, which is also in standard form (with a
= 1).

If one of a or b is zero, we know how to graph the equation and how to read off an equation from a graph. You probably suspect there will be some cases where it won't be so easy, and neither a nor b will be zero. You suspect right.

Okay, now what if an equation throws us a curveball? Should we sacrifice our bodies and take our base?

If neither a nor b is zero, we can most easily graph the linear equation by finding its intercepts.

Sample Problem

Graph the linear equation x + 4y = 8.

Let's find the intercepts. To find the x-intercept, let y = 0, since the x intercept will be at a point of the form (something, 0). Then

x + 4(0) = 8

and so x = 8 is the x-intercept.

For the y-intercept, let x = 0. Then

0 + 4y = 8

and y = 2 is the y-intercept. We have now determined both intercepts. Who needs a or b to be zero? Not us.

Now we can plot the intercepts:

and connect the dots to get the line:


Sample Problem

Write, in standard form, the linear equation graphed below:

The x intercept is -1, which means whatever a, b, and c are,

a(-1) + b(0) = c.

Let's make life easy on ourselves and let a = 1. That's right...we're going to dip this equation in a bucket of A-1 sauce.

(-1) = c.

To find b, the remaining coefficient, we look at the y-intercept: y = -2. x will be 0, and we have already decided that c = -1, so we find

0 + b(-2) = -1.

Therefore, . We now know all the coefficients and can write the equation

If we want to make things pretty, we can multiply both sides of the equation by 2 and write the resulting equation, which has integer coefficients. If we want to make things really pretty, we can dress the equation up in a sequined ball gown and give it a makeover. Let's start small, though:

2x + y = -2.

Sample Problem

Write, in standard form, the linear equation graphed below:

The x intercept is -2, which means whatever a, b, and c are,

a(-2) + b(0) = c.

We can let a = 1, so

(-2) = c.

To find b we look at the y-intercept, which occurs at y = 4. At the y-intercept x = 0, and since we've decided c = -2, we find

0 + b(4) = -2.

This means . We now know all the coefficients. Not on a first-name basis, but well enough to get by. We can now write the equation.

.

To make things pretty, we can multiply both sides of the equation by 2 to get an equivalent equation with integer coefficients:

2x -y = -4

Now for that makeover.

Standard Form Practice:

Example 1

Graph the linear equation 3x + 4y = 5.


Exercise 1

Graph the following linear equation: 3xy = 7.


Exercise 2

Graph the following linear equation: x + 2y = -5.


Exercise 3

Graph the following linear equation: -3x + 2y = 8.


Exercise 4

Determine the linear equation on the following graph:


Exercise 5

Determine the linear equation on the following graph:


Exercise 6

Determine the linear equation on the following graph:


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