Systems of two linear equations in two variables

`x = 8 x + 3y = 26`

The solution to the given system of equations is `(x, y)`. What is the value of `y`?

Answer: `6`

Substitute the value of `x` from the first equation `(x = 8)` into the second equation: `8 + 3y = 26`. Subtract `8` from both sides to get `3y = 18`. Finally, divide by `3` to find that `y = 6`.

At how many points do the graphs of the equations `y=x+20` and `y= 8x` intersect in the `xy`-plane?

A) `0`

B) `1`

C) `2`

D) `8`

Answer: B

Each given equation is written in slope-intercept form, `y = mx + b`, where `m` is the slope and `(0, b)` is the `y`-intercept of the graph of the equation in the `xy`-plane. The graphs of two lines that have different slopes will intersect at exactly one point. The graph of the first equation is a line with slope `1`. The graph of the second equation is a line with slope `8`. Since the graphs are lines with different slopes, they will intersect at exactly one point.

Note that choice A is incorrect because two graphs of linear equations have no intersection points only if they are parallel and therefore have the same slope. On the other hand, choice C and choice D incorrect because two graphs of linear equations in the `xy`-plane can have only `0`, `1`, or infinitely many points of intersection.

`6x + 7y = 28`

`2x + 2y = 10`

The solution to the given system of equations is `(x, y)`. What is the value of `y`?

A) –2

B) 7

C) 14

D) 18

Answer: A

The given system of linear equations can be solved by the elimination method. Multiplying each side of the second equation in the given system by `3` yields `2x+2y= 10(3)`, or `6x+6y = 30`. Subtracting this equation from the first equation in the given system yields: `6x+7y-(6x+6y)= (28) - (30)`, which is equivalent to `(6x−6x)+(7y-6y) = -2`, or `y= -2y = -2`.

`(x − 2) − 4(y + 7) = 117`

`(x − 2) + 4(y + 7) = 442`

The solution to the given system of equations is `(x, y)`. What is the value of `6(x − 2)`?

Answer: 1677

We use elimination for this system. Add the two equations together:
`(x − 2) − 4(y + 7) + (x − 2) + 4(y + 7) = 117 + 442`
The `4(y + 7)` terms cancel out, leaving `2(x − 2) = 559`.
The question asks for the value of `6(x − 2)`. To get this, multiply both sides of `2(x − 2) = 559` by 3, which results in `6(x − 2) = 1677`.

`48x − 64y = 48y + 24 ry = 1/8 − 12x`

In the given system of equations, `r` is a constant. If the system has no solution, what is the value of `r`?

Answer: `-28`

A system of linear equations has no solution when the lines are parallel and have different `y`-intercepts. First, write both equations in the standard form `Ax + By = C`.

Equation 1: `48x - 64y - 48y = 24 → 48x - 112y = 24`

Equation 2: `12x + ry = 1/8`

For the lines to be parallel, the ratio of the `x`-coefficients must equal the ratio of the `y`-coefficients. The `x`-coefficient in the first equation `(48)` is `4` times the `x`-coefficient in the second equation `(12)`.

Therefore, the `y`-coefficient in the first equation `(-112)` must also be `4` times the `y`-coefficient in the second equation `(r)`. `-112 = 4 xx r r = -112 / 4 = -28`.