Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given y = 4x + 4 and 3x + 2y = 12?

Answer 1

Here's what you could do.

Explanation:

Your starting equations are

`{ (y = 4x + 4), (3x + 2y = 12) :}`

Now, you can either solve this system and see if you get an unique pair of value for `x` and `y`, which would of course imply that the system of equations has one solution.

Alternatively, you can write these equations in slope-intercept form and compare their slopes and `y`-intercepts.

Let's say you want to use both methods.

• Find the two values of `x` and `y` by solving the system

The first thing to do here is rearrange them so that the `y` and `x`-terms are on the same side of the equations

`{(-4x + y = color(white)(x)4), (color(white)(x)3x + 2y = 12) :}`

At this point, you have some options to chose from. You can use the substitution method or the elimination method. I'll use the latter.

Notice that if you multiply the first equation by `(-2)` and add the left-hand sides and right-hand sides of the two equations separately, you can eiminate the `y` term.

This will allow you to find the value of `x`.

`{(-4x + y = color(white)(x)4 | xx (-2)), (color(white)(x)3x + 2y = 12) :}`

`{(color(white)(x)8x -2y = -8), (color(white)(x)3x + 2y = 12) :}`
`color(white)(x)stackrel("-------------------------------------")`

`8x - color(red)(cancel(color(black)(-2y))) + 3x + color(red)(cancel(color(black)(-2y))) = -8 + 12`

`11x = 4 implies x = 4/11`

This means that `y` will be

`y = 4 * (4/11) + 4 = 60/11`

Since solving the system produces an unique value of `x` and an unique value of `y`, it follows that the system has one solution.

• Use slope-intercept form

The slope-intercept form of a line is given by the equation

`color(blue)(y = mx + b)" "`, where

`m` - the slope of the line;
`b` - the `y`-intercept.

In your case, the first equation is already written in slope-intercept form

`y = 4x + 4 implies {(m = 4), (b = 4) :}`

The second equation's slope-intercept form will be

`2y = -3x + 12`

`y = -3/2x + 6 implies {(m = -3/2), (b = 6) :}`

Because the two lines have different slopes, it follows that they will intersect in one point. Once again, this implies that the system of equations has one solution.

graph{(4x - y + 4)(3x + 2y - 12) = 0 [-10, 10, -2, 10]}