What do you think the graph of $4x+2y=4$ looks like? In this lesson you will learn about the
graph of this and other similar equations.
The slope of a line is a measurement of how steep it is. Use the figures
below to help remind you about how to measure the slope of a line. The
first line graphed below has slope $$3/2$$, while the second one has slope $$3/2$$.
Look at the grid to the left to answer the following questions.
What is the slope of $4x+2y=4$?
 
What is the $y$intercept of $4x+2y=4$ (where does it
cross the $y$axis)?
 
What is the $x$intercept of $4x+2y=4$ (where does it
cross the $x$axis)?
 
The effects of A, B, and C
To the left is the graph of $Ax+By=C$.
You are looking at the graph of $4x+2y=4$
(because $A=4$, $B=2$, $C=4$). Use the sliders to change the values of $A$, $B$, and $C$ and
notice how each affects the shape and position of the graph of
$Ax+By=C$.
Each row of the following table has a question about how the graph changes when you change
$A$, $B$, or $C$. Answer the question for each of the three variables.
 When $A$ changes  When $B$ changes  When $C$ changes

Does the graph of the equation remain a line?




Does the $x$intercept of the graph change?




Does the $y$intercept of the graph change?




Does the slope of the graph change?




The form $Ax+By=C$, where $A$, $B$, and $C$ are
constants, is called the standard form for
an equation for a line.
Each row of the table below gives an equation for a line in standard form.
Using the sliders, change the values of $A$, $B$, and $C$, and complete this table. In each
case, find the slope of the line by looking at the grid to the left.
Using the pattern in the table, what do you think the slope of
$2x+5y=40$ is?
 
The slopes you have found suggest that:
The slope of the line $Ax+By=C$ is $$A/B$$.
Converting from standard form to slopeintercept form
In the last question, you found the slopes of lines by looking at their graphs. If you want
to find the slope of a line algebraically, by using its
equation, you can convert that equation into slopeintercept form.
For example, we can write the equation $4x+2y=4$ in slopeintercept form as follows:
$$
\cl"tight majoinalign"{\table
\text"Standard form: ", , 4x+, 2y, =, , 4;
\text"Subtract" \: 4x \: \text"from both sides: ", 4x+, 4x+, 2y, =, 4x+, 4;
, , , 2y, =, 4x+, 4;
\text"Multiply both sides by" \: 1/2 \text": ", , , 1/2(2y), =, \colspan 2{1/2(4x+4)};
\text"Distribute: ", , , y, =, 1/2(4x)+, 1/2(4);
\text"Slopeintercept form: ", , , y, =, 2x+, 2
}
$$
Each row of the table has an equation in standard form. Write that
equation in slopeintercept form.
Finding the $x$ and $y$ intercepts of a line in standard form
The $y$intercept of a line is the point on that line where $x=0$, and the $x$intercept is
the point where $y=0$. You can use this to find the intercepts of lines algebraically.
Algebraically find the $y$intercept of each line in the table below.
 equation for the line  set $x=0$  $y$intercept
