Equations for Lines in Standard Form

What do you think the graph of $4x+2y=4$ looks like? In this lab you will learn about the graph of this and other similar equations.

Question 1. On the grid to the left you can see the graph of $4x+2y=4$. What does the graph of $4x+2y=4$ look like?

Question 2. 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.

Graph Graph

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

Question 3. Click on to see 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 $Ax+By=C$.

What happens to the graph when $A$ changes?

What happens to the graph when $B$ changes?

What happens to the graph when $C$ changes?

Question 4. Which of $A$, $B$, $C$ affect the $x$-intercept?

Which of $A$, $B$, $C$ affect the $y$-intercept?

Which of $A$, $B$, $C$ affect the slope?

Question 5. Use the sliders to set $A=4$, $B=2$, and $C=4$. You are looking at the graph of $4x+2y=4$. Slide the value of $A$ toward zero. You can use the arrow keys to make small adjustments.

What happens to the slope of the graph as $A$ gets closer and closer to zero? Does the graph become steeper or flatter?

What is the slope of the graph when $A=0$?

Question 6. Set $A$ back to 4 and slide the value of $B$ toward zero. What happens to the slope of the graph as $B$ gets smaller? Does the graph become steeper or flatter?

Question 7. Using the sliders, change the values of $A$, $B$, and $C$, and complete this table. In each case, find the slope by looking at the grid to the left.

$Ax+By=C$$A$$B$slope
$2x+3y=6$23−2/3
$x+2y=2$
$3x+2y=6$
$-2x+3y=6$

What is the relationship between $A$, $B$, and the slope of the line?

Using the relationship you found, what do you think the slope of $2x+5y=10$ is? Check your answer using the sliders.

The slope of the line $Ax+By=C$ is $-A/B$.

Look at the table you completed in Question 7. Are the slopes you found the same as $-A/B$? If not, look at each graph again and make sure you calculated the slope correctly.

Finding an equation for a line in Standard Form

The form $Ax+By=C$, where $A$, $B$, and $C$ are constants, is called the standard form for an equation for a line.

Question 8. Use the sliders to change the values of $A$, $B$, and $C$, and answer the following questions.

Find an equation (in standard form) for the line with $x$-intercept $(2,0)$ and slope 3.
For the equation you just found, find the slope using the formula $m=-A/B$. Show the values of $A$ and $B$.
$A=$
$B=$
$m=$
Question 9. Find an equation (in standard form) for the line with $x$-intercept $(-2,0)$ and $y$-intercept $(0,1)$.

What is the slope of this line?

Now, find the slope of this line using the formula $m=-A/B$. (Should this answer be the same as your answer to the previous question?) Show the values of $A$ and $B$.
$A=$
$B=$
$m=$

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