Math CentralQuandaries & Queries


Question from Mary, a student:

Having problems doing this problem, looking for a solution with the work. I would like to see how you got your answer, to see what I was doing wrong.

solve using the substitution method, is there "no solution" or "infinitely many solutions"


We have two responses for you

Hi Mary.

Two equations could be in any one of three relationships to each other:

  1. They are different expressions of exactly the same line. For example, y=2x and 2y = 4x are actually the same line. In this case, there are "infinitely many solutions" because there are an infinite number of values of x that give a value for y that matches in both equations. Notice as well that when this is the case, the slopes and the y-intercepts of the two equations would match.
  2. They are parallel lines. For example, y = 2x and y = 2x + 1 are parallel. Parallel lines have the same slope, but different y intercepts. Since there are no points (x, y) that simultaneously are on both lines, we say there is "no solution".
  3. They intersect in one point. The one point is the "unique solution". This can only be the case if the two equations have different slopes. The y intercept doesn't matter.

To determine which of these cases you have for a given pair of equations, often the easiest thing to do is write both equations in y = mx + b form and compare the slopes, then if needed, compare the y intercepts. This won't tell you what the unique solution actually is, however (if they intersect in a single point, rather than being identical or parallel lines).

Here's how things play out when you use the substitution method to "solve" two equations:

Example 1:

2x + y = 1
-3x + 2y = 0

Solve one equation for a single variable (which one? just pick what looks easiest!)
    2x + y = 1
    y = 1 - 2x

Then substitute this expression (which equals y) in for y in the other equation. So
    -3x + 2y = 0
    -3x + 2(1 - 2x) = 0

And solve for x:
    -3x + 2 - 4x = 0
    -7x = -2
    x = 2/7.

Since it gave us a single value of x, I know that we will get a unique solution. I use this value of x to find the value of y. Just choose one of the original equations (doesn't matter which one) and substitute 2/7 in for x.
    2x + y = 1
    2(2/7) + y = 1
    y = 3/7.

So the unique solution to this pair of equations is (2/7, 3/7).

Let's look at the other two situations to see what would have happened.

Example 2:

2x + y = 1
-2x - y = 2

Solve the first for y:
    y = 1 - 2x

Substitute into second equation:
    -2x - (1 - 2x) = 2
    -2x - 1 + 2x = 2
    -1 = 2.

That's a contradiction, obviously! So that means there is no solution. These two equations are parallel lines.

Example 3:

2x + y = 1
6x + 3y = 3

Solve the first for y:
    y = 1 - 2x

Substitute into the second equation:
    6x + 3(1 - 2x) = 3
    6x + 3 - 6x = 3
    3 = 3.

This is a truism: it is true regardless of the value of x, so there are an infinite number of solutions. These two equations are actually just two ways of expressing the same equation (multiply the first equation by 3 on both sides and you'll verify this).

I hope this explanation and set of examples helps you solve any problems you have with two linear equations.

Stephen La Rocque.


Hi Mary,

Without actually solving this system of equations we can determine that there will in fact be ONLY ONE solution. The first equation has a slope of -4 (we can rearrange it to read y=-4x+4) and the second equation has a slope of -1/4 (we can rearrange it to read y=-1/4x). When two lines have different slopes, they are guaranteed to intersect at one point, giving us one solution.

If this system were to have no solutions or infinitely many solutions, the equations would have to have the same slope. Additionally, they would have to have different y-intercepts to have no solution and the same y-intercept to have infinitely many solutions.

Hope this helps.


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