Solve the system of equations. y = -28x - 2 y = x^2 - 14x + 47 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______)

Question

Solve the system of equations. y = -28x - 2 y = x^2 - 14x + 47    Write the coordinates in exact form. Simplify all fractions and radicals. (______,______)

Bytelearn · Accepted Answer

Substitute y: Substitute y from the first equation into the second equation. Since y = -28x - 2, we can replace y in the second equation with -28x - 2. The second equation becomes -28x - 2 = x^2 - 14x + 47.Solve for x: Now, we need to solve for x. To do this, we will set the equation to zero by moving all terms to one side: x^2 - 14x + 47 = 28x + 2. This simplifies to x^2 - 42x + 45 = 0.Factor the quadratic equation: Next, we factor the quadratic equation x^2 - 42x + 45 = 0. The factors of 45 that add up to 42 are 1 and 45, but we need them to subtract to 42, so the factors are 3 and 15. This gives us (x - 3)(x - 15) = 0.Find solutions for x: Setting each factor equal to zero gives us the solutions for x. So, x - 3 = 0 or x - 15 = 0. This means x = 3 or x = 15.Substitute x back: Now we substitute x back into the first equation y = -28x - 2 to find the corresponding y values. First, for x = 3, we get y = -28(3) - 2, which simplifies to y = -84 - 2, so y = -86.Find corresponding y values: Next, for x = 15, we substitute into the first equation y = -28x - 2 to get y = -28(15) - 2, which simplifies to y = -420 - 2, so y = -422.Identify solutions: We now have two solutions for the system of equations: (3, -86) and (15, -422). These are the coordinates of the points where the two equations intersect.

Solve the system of equations. $\newline$ $y = -28x - 2$ $\newline$ $y = x^2 - 14x + 47$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,)

Full solution

More problems from Solve a nonlinear system of equations

Related topics

Solve the system of equations.\newliney=−28x−2y = -28x - 2y=−28x−2\newliney=x2−14x+47y = x^2 - 14x + 47y=x2−14x+47\newlineWrite the coordinates in exact form. Simplify all fractions and radicals.\newline(______,______)

Solve the system of equations. $\newline$ $y = -28x - 2$ $\newline$ $y = x^2 - 14x + 47$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,)