Question

@mathmale

Answer 1

An equation is shown below.

2x + y = 3

Part A: Explain how you will show all of the solutions that satisfy this equation. (4 points)

Part B: Determine three different solutions for this equation. (4 points)

Part C: Write an equation that can be paired with the given equation in order to form a system of equations that is inconsistent. (2 points)

Answer 2

@mathmale

Answer 3

Hi, DG: I'd suggest you actually graph that equation; the graph just might help you answer the question.

Answer 4

Actually, DG, the graph itself represents all of the solutions to the given equation. Any graph is made up of infinitely many solutions, all jammed together on a line or curve.

Answer 5

Part A: To show "all" solutions, you could solve for y giving you y = -2x + 3
This essentially saying that, in 2D space (cartesian plane), any point with the coordinates (x, -2x+3) solves that solution.

We can show this solution as with a graph. In this case, all the points to your solution form a line with a y-intercept of 3 (implies x= 0) and a slope of -2.

B. Choose any three points on that line and you get three solutions! Or just plug in 3 different x values and solve for y. Ex:

Let x = 4
y = -2(4) + 3 = -8+3 = -5
(4,-5)

C. A system of equations is "inconsistent" if no solutions exist. Graphically, this means that two lines are not intersecting. Can you draw a line that does not intersect that graph of our above equatioN?

Answer 6

One way of approaching part b would be to solve 2x+y=3 for y, and then choosing any 3 x values, and then evaluting your equation for y at each of those x values.

Answer 7

Part C: I'd like to hear your interpretation of this question first.

Answer 8

DG: @kmeis002 has explained Part C very clearly and well. I'd suggest you follow his advice.

Answer 9

Okay. I am back. :3 I am reading the stuff now.

Answer 10

Thank you both. I understand clearly now. I thought only two lines can have a solution/solutions!

Answer 11

You're welcome; come back soon. Regarding your last sentence: "I'd thought that two intersecting lines could have a unique solution and that two congruent lines infinitely many solutions." Two lines that do not intersect represent "inconsistent" equations; there's no solution.

Answer 12

O.K.