Question

Is there an ordered pair that is a solution to BOTH of these linear equations?

*You can type them into the Desmos calculator to see the graph.



`y=x+1`

`y=-x+5`

Answer 1

To determine if there is an ordered pair that is a solution to both linear equations \( y = x + 1 \) and \( y = -x + 5 \), we can set the right-hand sides of the equations equal to each other and solve for \( x \):

\[
x + 1 = -x + 5
\]

Combining like terms, we add \( x \) to both sides:

\[
x + x + 1 = 5
\]

This simplifies to:

\[
2x + 1 = 5
\]

Next, we subtract 1 from both sides:

\[
2x = 4
\]

Now, divide both sides by 2:

\[
x = 2
\]

Now that we have \( x = 2 \), we can substitute this value back into either of the original equations to find the corresponding \( y \)-value. Let's use the first equation \( y = x + 1 \):

\[
y = 2 + 1 = 3
\]

So the ordered pair that is a solution to both equations is:

\[
(2, 3)
\]

To verify, we can substitute \( x = 2 \) into the second equation \( y = -x + 5 \):

\[
y = -2 + 5 = 3
\]

Thus, \( (2, 3) \) satisfies both equations.

Therefore, the solution is:

\[
\boxed{(2, 3)}
\]