Question

Explain how you can determine that the following system has one unique solution – without actually solving the system.

2x+y=4
2y=6-2x

Answer 1

To do so, get every part into slope-intercept form (y = mx + b).

If the slopes are different, there is one solution.

If the slopes the same but the y intercepts different, there is no solution.

If the slopes and y intercepts are the same, there are infinitely many solutions.

2x + y = 4

2y = 6 - 2x

Solve for y on both

y = 4 - 2x

y = 3 - x

They are both different, so there is one solution

Answer 2

We want to see how we can determine if a system of equations has only one solution (without solving the system).

Remember that the solution on a system of equations is the point where the graphs of the equations intersect. So the system has only one solution if the equations intersect only once.

Because here we have two linear equations, we need to check that these lines are not parallel (parallel lines never intersect) and that these equations do not represent the same line (if both equations are the same line, then they intersect in infinite point).

The system is:

2x + y = 4

2y = 6 - 2x

Isolating y on both equations leads to:

y = -2x + 4

y = (6 - 2x)/2 = -x + 3

We can see that both equations have different slopes, meaning that these equations are not parallel, and this means that the lines must intersect at some point. Then we have a unique solution for the system.

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