The graph shows the parabola $y=-(x+4)^2+5$ . Consider the following linear equation: $y=-3x+b$ for some constant $b$ . If one of the solutions to the system of equations formed by the parabola and the linear equation is $(-4,5)$ , which of the following is the other solution?

View step-by-step help

Home

Math Problems

Precalculus

Convert equations of conic sections from general to standard form

Full solution

Q. The graph shows the parabola

y=-(x+4)^2+5

. Consider the following linear equation:

y=-3x+b

for some constant

b

. If one of the solutions to the system of equations formed by the parabola and the linear equation is

(-4,5)

, which of the following is the other solution?

Find $b$ in linear equation: First, we need to find the value of $b$ in the linear equation $y=-3x+b$ using the given solution $(-4,5)$ . $\newline$ Substitute $x$ with $-4$ and $y$ with $5$ into the linear equation. $\newline$ $5 = -3(-4) + b$ $\newline$ $5 = 12 + b$ $\newline$ Now, solve for $b$ . $\newline$ $b$ $1$ $\newline$ $b$ $2$
Set equations equal: Now that we have the linear equation $y=-3x-7$ , we can set it equal to the parabola equation $y=-(x+4)^2+5$ to find the other point of intersection. $-(x+4)^2+5 = -3x-7$
Expand and simplify: Next, we need to expand the left side of the equation and simplify. $\newline$ $-x^2 - 8x - 16 + 5 = -3x - 7$ $\newline$ $-x^2 - 8x - 11 = -3x - 7$
Move terms and solve: Now, we will move all terms to one side to set the equation to zero and solve for $x$ . $-x^2 - 8x + 3x - 11 + 7 = 0$ $-x^2 - 5x - 4 = 0$
Factor quadratic equation: We can solve the quadratic equation $-x^2 - 5x - 4 = 0$ by factoring or using the quadratic formula. However, since we already know one solution is $x = -4$ , we can factor by grouping. $\newline$ $x = -4$ is a solution, so $(x + 4)$ is a factor. $\newline$ Let's factor $-x^2 - 5x - 4$ . $\newline$ $-x^2 - 4x - x - 4 = 0$ $\newline$ $-x(x + 4) - 1(x + 4) = 0$ $\newline$ $(x + 4)(-x - 1) = 0$
Find y-coordinate: The solutions to the equation $(x + 4)(-x - 1) = 0$ are $x = -4$ and $x = -1$ . We already know the solution $(-4,5)$ , so we need to find the y-coordinate when $x = -1$ . Substitute $x = -1$ into the linear equation $y = -3x - 7$ . $y = -3(-1) - 7$ $y = 3 - 7$ $y = -4$
Final solution: The other solution to the system of equations is $(-1,-4)$ .