Question

The line $y=mx+2$ intersects the parabola $y=ax^{2}+5x-2$ at $(1,5)$. Then the value of $a+m$ is equal to

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

Algebra Tip: "Intersection" is just a fancy word for "a point they both share." Whenever you are given a point that lies on a curve, immediately plug those $x$ and $y$ coordinates into the equation!

Answer 1

Answer: (E)

Concept:
When two graphs intersect at a specific point $(x, y)$, it means that the coordinates of that point must perfectly satisfy the equations of both graphs simultaneously. By substituting the intersection point into each equation independently, we can solve for any unknown coefficients.

Step 1: Identify the point of intersection.
We are given that both the line and the parabola pass through the point $(1, 5)$. This means we can substitute $x = 1$ and $y = 5$ into both equations.

Step 2: Substitute the point into the line equation.
The equation of the line is $y = mx + 2$. Substitute $(1, 5)$: $$5 = m(1) + 2$$

Step 3: Solve for the slope m.
Subtract 2 from both sides to isolate $m$: $$m = 5 - 2$$ $$m = 3$$

Step 4: Substitute the point into the parabola equation.
The equation of the parabola is $y = ax^2 + 5x - 2$. Substitute $(1, 5)$: $$5 = a(1)^2 + 5(1) - 2$$ $$5 = a + 5 - 2$$ $$5 = a + 3$$

Step 5: Solve for a and calculate the final sum.
Subtract 3 from both sides to find $a$: $$a = 5 - 3 = 2$$ The question asks for the value of $a + m$: $$a + m = 2 + 3 = 5$$ Hence the correct answer is (E) 5.