Graph of ax+by=c: How a,b,c Affects Graph

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In summary, the conversation discusses the effects of a, b, and c on the equation ax+by=c graphically. It suggests rewriting the equation in matrix form and using Desmos to find the coordinates and distance of the intersection point. It also explores the different scenarios when a, b, and c have different values and their impact on the equation's graph.
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How does a,b,c effects the equation ax+by=c graphically. Thanks!
 
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sahilmm15 said:
How does a,b,c effects the equation ax+by=c graphically. Thanks!
Write it as ##\begin{bmatrix} a & b \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix}=c ## and ask again. What does ##c=0 ## and ##c\neq 0## mean, and what stands ##(a,b)## for?
 
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Go to www.desmos.com/calculator/
Type in
ax+by=c
ax=c
by=c
(a,b)
y=(b/a)x

Find the x-coordinate of the intersection
by plugging in the last equation into the first equation and solve for x.

Solve the y=(b/a)x for x.
Plug that new equation into the first equation and solve for y

How far is that point from the origin? Call it d.
Enter another equation
r=d
 
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sahilmm15 said:
How does a,b,c effects the equation ax+by=c graphically. Thanks!
If a = 0 and b = 0, the equation is 0x + 0y = c. What can you say about c? Does this equation have any solutions?
If b = 0, and ##a \ne 0##, the equation is ax = c, or ##x = \frac c a##. What does the graph of this equation look like?

If a = 0 and ##b \ne 0##, the equation is by = c, or ##y = \frac c b##. What does the graph of this equation look like?

If ##a \ne 0## and ##b \ne 0## solve for y in terms of x and the constant c. What does the graph of this equation look like?
 
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FAQ: Graph of ax+by=c: How a,b,c Affects Graph

How does the value of a affect the graph of ax+by=c?

The value of a in the equation ax+by=c affects the slope of the graph. A larger value of a will result in a steeper slope, while a smaller value of a will result in a flatter slope. If a is negative, the graph will be reflected across the x-axis.

What is the significance of the value of b in the equation ax+by=c?

The value of b in the equation ax+by=c affects the y-intercept of the graph. A larger value of b will result in a higher y-intercept, while a smaller value of b will result in a lower y-intercept. If b is negative, the graph will be reflected across the y-axis.

How does the constant term c impact the graph of ax+by=c?

The constant term c in the equation ax+by=c affects the position of the graph on the coordinate plane. It represents the point where the graph intersects the y-axis when x=0. So, a larger value of c will shift the graph up, while a smaller value of c will shift the graph down.

What happens to the graph when all three variables, a, b, and c, are changed?

When all three variables, a, b, and c, are changed, the graph will undergo a transformation. The slope, y-intercept, and position on the coordinate plane will all be affected. For example, changing a and b while keeping c constant will result in a different slope and y-intercept, but the graph will still intersect the y-axis at the same point.

Can the graph of ax+by=c be a straight line?

Yes, the graph of ax+by=c can be a straight line. This will occur when either a or b is equal to 0, resulting in a horizontal or vertical line, respectively. It can also be a straight line when a and b are both non-zero, but their ratio is constant.

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