Given the equations of two lines $y=2x+1$ and $y=-x+4$, determine whether the lines intersect. If they do, find the point of intersection.

• A. Yes, the lines intersect at $(1,3)$
• B. No, the lines are parallel and do not intersect.
• C. Yes, the lines intersect at $(2,5)$
• D. Yes, the lines intersect at $(0,1)$

Answer: (A)

Verified step by step guidance

1. Identify the two given lines: Line 1 is \(y = 2x + 1\) and Line 2 is \(y = -x + 4\).
2. To find if the lines intersect, set their right-hand sides equal to each other because at the point of intersection, both lines have the same \(x\) and \(y\) values. So, solve the equation \$2x + 1 = -x + 4$.
3. Rearrange the equation to isolate \(x\): add \(x\) to both sides and subtract 1 from both sides to get \$3x = 3$.
4. Solve for \(x\) by dividing both sides by 3, which gives \(x = 1\).
5. Substitute \(x = 1\) back into either original equation to find the corresponding \(y\) value. For example, using \(y = 2x + 1\), calculate \(y = 2(1) + 1\).