Today’s Question of the Day:

How many solutions can you find for a system of linear equations?

Case 1: Infinitely Many Solutions

Solve for x and y. y = 2x – 1
2y = 4x – 2


Answer:
Upon first comparison, the equations look different. However, if we divide the second equation by 2, we get
y = 2x - 1, which is equal to the first equation.

Since the equations are equal, the slopes and y-intercepts are equal. Let’s now look at the values of x and y on the two lines.

Take x = 1, for example.
Solving for y in the first equation, y = 2(1) - 1 = 1, so y = 1.
Solving for y in the second equation, 2y = 4(1) - 2 = 2, so 2y = 2 and y = 1.

So y = 1 when x = 1, regardless of what equation is used. If we test any other x, we will find that the y values are also the same. As a result, there are infinitely many solutions to this system of linear equations.

When looking at the graph, it looks like there is only one line. This is because the graphs are overlapping each other. Every point on the first line is equal to the point on the second line!

Case 2: No Solution

Solve for x and y. y = 2x – 1
y = 2x + 3


Answer:
We notice that the two equations have the same slope, but different y-intercepts:
Equation 1: slope = 2, y-intercept = -1
Equation 2: slope = 2, y-intercept = 3

Since equations that have the same slope and different y-intercepts will never intersect, there is no solution.

When can also find the answer by looking at the graph. We see that the lines never cross. Since there are no intersections, there are no solutions.

Case 3: One Solution

Solve for x and y. y = 2x – 1
y = -1/2 x + 4


Answer:
If we look at the graph, we see that there is 1 solution. The lines cross each other. Though it is hard to tell what the exact solution is from the graph, we can take a guess. Take your best guess and write it down. We’ll check it at the end!

The easiest way to solve this equation is by using substitution. We do this by setting y from the first equation to y in the second equation.

y = 2x - 1 = -1/2 x + 4

If you’re like me, you don’t like working in fractions. The great thing about equations is that you can manipulate numbers by adding, subtracting, multiplying, or dividing to both sides of the equation. Here, we can multiply both sides of the equation by 2.

2(2x - 1) = 2(-1/2 x + 4)

Simplify by distributing the 2.
4x - 2 = -x + 8

And combine like terms.
5x = 10

Solving for x, we get x = 2. Plugging this into either equation, we find that y = 3.
y = 2(2) - 1 = 3, so y = 3.

There is, therefore, one solution to this system of equations: x = 2, y = 3. Is this the same answer you guessed from the graph?

Re-cap:

1. A system of linear equations that have the same slope and same y-intercept has infinitely many solutions.
2. A system of linear equations that have the same slope and different y-intercepts has no solution.
3. A system of linear equations that have different slopes and different y-intercepts has one solution.

*Bonus question*
Solve for x and y. x = 3
y = 5

*Side Learning*
Equations that have the same slopes are parallel lines. The equations in Case 1 and 2 are thus parallel.

Now look at the slopes for the equations in Case 3:
Equation 1: slope = 2
Equation 2: slope = -1/2

When the slopes of equations are opposite, reciprocals of each other, then the lines are perpendicular.

Let’s try this for a slope of 2, as seen in equation 1!

The number 2 is a positive number and can be written as +2. For a number to be opposite, the number has to take on the opposite sign. In this case, the opposite sign is negative and the opposite of 2 is -2. The reciprocal of a number x, is 1 divided by x (or 1/x). For 2, the reciprocal is then 1/2.

Combining these two, the opposite reciprocal of 2 is -1/2. Notice that this is the same slope as seen in equation 2. We can see that the lines represented by equations 1 and 2 are perpendicular.