Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$?
Two lines that are parallel have the same slope. Therefore, the slope of line $a$ is $2$. Using the point-slope formula, we get that the equation for line $a$ is $y-5=2(x-2)=2x-4$. In slope-intercept form, the equation is $y=2x+1$. Therefore, the y-intercept is $\boxed{1}$.
1
Level 3
Algebra
Problem: Line $a$ is parallel to line $y=2x+4$ and passes through the point $(2,5)$. What is the y-intercept of line $a$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the y-intercept of line $a$, we first need to determine its slope. Since line $a$ is parallel to the line given by the equation $y=2x+4$, it shares the same slope. The slope of the line $y=2x+4$ is $2$. Therefore, the slope of line $a$ is also $2$.

Next, we can use the point-slope form of a line to find the equation of line $a$. The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Here, we have the point $(2,5)$ and the slope $m=2$.

Substituting these values into the point-slope form, we get:
y - 5 = 2(x - 2)

Now, we simplify this equation to find the slope-intercept form (y = mx + b):
y - 5 = 2x - 4
y = 2x + 1

In this equation, the y-intercept is the value of $b$, which is the constant term when the equation is expressed in the form $y = mx + b$. From our derived equation, we see that the y-intercept is $1$.

Thus, the final answer is $1$.<|END_OF_LATENT|> The final answer is $1$.