Question

1. $y = 2x + 3$ and $y = x + 5$

solution:

Explanation:

Step1: Set equations equal (intersection)

Since $y = 2x + 3$ and $y = x + 5$, set $2x + 3 = x + 5$

Step2: Solve for $x$

Subtract $x$ and 3 from both sides:
$2x - x = 5 - 3$
$x = 2$

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

Substitute $x=2$ into $y = x + 5$:
$y = 2 + 5 = 7$

Step4: Verify with second equation

Substitute $x=2$ into $y=2x+3$:
$y = 2(2) + 3 = 7$ (matches)

Answer:

The lines intersect at the point $(2, 7)$

To plot the lines:

1. For $y=2x+3$: plot the y-intercept $(0,3)$, then use slope $2$ (rise 2, run 1) to find another point like $(1,5)$.
2. For $y=x+5$: plot the y-intercept $(0,5)$, then use slope $1$ (rise 1, run 1) to find another point like $(1,6)$.
3. The two lines cross at $(2,7)$.