Question

1. which ordered pair is the solution to this system of equations?

y = x + 4
x + y = 2
options: (-1, 3), (1, 5), (0, 2), (-4, 0)

1. what is a solution for the system of equations x - y = 2 and y = 2x - 4?

options: (3, 2), (4, 2), (0, 2), (2, 0)

1. the system of equations \\(\

$$\begin{cases}x + 2y = 6 \\\\ 2x + 4y = 8\\end{cases}$$

\\) has:
options: no solution, two solutions, one solution, none of these

Explanation:

Problem 1

Step1: Substitute $y=x+4$ into second equation

$x + (x + 4) = 2$

Step2: Simplify to solve for $x$

$2x + 4 = 2 \implies 2x = -2 \implies x = -1$

Step3: Find $y$ using $x=-1$

$y = -1 + 4 = 3$

Problem 2

Step1: Substitute $y=2x-4$ into first equation

$x - (2x - 4) = 2$

Step2: Simplify to solve for $x$

$x - 2x + 4 = 2 \implies -x = -2 \implies x = 2$

Step3: Find $y$ using $x=2$

$y = 2(2) - 4 = 0$

Problem 3

Step1: Simplify the second equation

Divide $2x+4y=8$ by 2: $x + 2y = 4$

Step2: Compare to first equation

First equation: $x+2y=6$; the two equations represent parallel lines (same left-hand side, different constants) so there is no solution.

Answer:

1. $(-1, 3)$
2. $(2, 0)$
3. no solution