Graphing system of equations

In summary, to solve the system of equations graphically, you can use the method of substitution to find the intersecting points of the two equations. This can then be checked algebraically by plugging in the coordinates of the intersecting points into each equation and verifying that they satisfy both equations.
  • #1

Homework Statement

Solve the system of equations graphically.

Homework Equations

4x - y = 5
y = 4 - 5x^2

2x^2 + y^2 = 33
x^2 - y^2 = 12

The Attempt at a Solution

The answers I got for the intersecting points are:
(1, -1)


(+/- 3.7, +/- 1.8)

These are estimates from graphing, but I'm not sure if I'm close... I don't know how to check myself algebraically yet... =/
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  • #2
You should be able to at least substitute your graphically-obtained points into the equations.
  • #3
I did that mostly...

but for the equations like y = 4 - 5x^2, is that the same as saying:

y = 4 - 5(-1)^2?
or would it bet
y = 4 - 5(-1^2)?

for that particular equations here are my points:

x...| y
0 ....4
+/-.89 ...0
+/- 1 ...-1
+/- 2 ...-3
  • #4
for your first problem you have found a possible solution at (1,-1).

Your equations are:
y= 4x -5
y= 4 - 5 x 2

Plugging x = 1 into each of these yields:

y = 4(1) -5 = -1

Thus your point is a solution for this equation.

Repeat for the second equation.

y = 4 - 5(-1) 2 = 4 -5 = -1

Thus your solution works in both equations and is an intersection point.

Do the same thing with your second problem.
  • #5
groovy, good to know I'm on the right path...

could you show me how to solve algebraically (#1)? If you could give me a start I'll work on it and let you know where I get...
  • #6
In my last post I expressed both equatons of your first problem as y expressed in terms of x. Eliminate y by setting them equal, then solve for x.

y= 4x -5
y = 4 - 5x 2
4x -5 = 4 - 5x 2
5x 2 + 4x - 9 = 0

Can you finish?

What is a graphing system of equations?

A graphing system of equations is a set of two or more equations that are graphed on the same coordinate plane. The solutions to the system of equations are the points where the graphs intersect.

How do you graph a system of equations?

To graph a system of equations, you must first solve each equation for y in terms of x. Then, plot the points from each equation on the same coordinate plane. The solution to the system of equations will be the point where the graphs intersect.

What do the points of intersection represent in a graphing system of equations?

The points of intersection represent the solutions to the system of equations. In other words, they are the values of x and y that make both equations true at the same time.

Can a graphing system of equations have more than one solution?

Yes, a graphing system of equations can have more than one solution. This is possible when the two equations have different slopes and intersect at more than one point on the coordinate plane.

What is the significance of a parallel or perpendicular lines in a graphing system of equations?

If the two equations in a graphing system have the same slope, they will never intersect and the system will have no solution. If the two equations have slopes that are negative reciprocals, they will intersect at a right angle and the system will have exactly one solution.

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