Question

solve the system of equations: $y = 3x - 20$ and $y = -x^2 + 34$.
a. $(6, -2)$ and $(-9, -47)$
b. no solution
c. $(9, -47)$
d. $(-2, 6)$

what is the solution to the system $y = -\frac{1}{2}x + 2$ and $y = x^2 - 6x + 8$?
a. no solution
b. $(1.5, 1.25)$
c. $(-2.5, 6)$
d. $(1.5, 1.25)$ and $(4, 0)$

if the solution to a system of equations is $x = 4, y = 1$, what do these coordinates represent in the context of graphing?
a. the slope of the line
b. the point where the line and parabola intersect
c. the y-intercept of the line
d. the vertex of the parabola

Explanation:

First Sub - Question (Solve the system \(y = 3x - 20\) and \(y=-x^{2}+34\))

Step 1: Set the equations equal

Since both equations are equal to \(y\), we set \(3x - 20=-x^{2}+34\).

Step 2: Rearrange into standard quadratic form

Rearrange the equation to \(x^{2}+3x - 20 - 34 = 0\), which simplifies to \(x^{2}+3x - 54 = 0\).

Step 3: Factor the quadratic

Factor the quadratic equation: \(x^{2}+3x - 54=(x + 9)(x - 6)=0\).

Step 4: Solve for \(x\)

Set each factor equal to zero: \(x+9 = 0\) gives \(x=-9\); \(x - 6=0\) gives \(x = 6\).

Step 5: Find the corresponding \(y\) values

For \(x = 6\), substitute into \(y = 3x-20\): \(y=3\times6 - 20=18 - 20=-2\).
For \(x=-9\), substitute into \(y = 3x - 20\): \(y=3\times(-9)-20=-27 - 20=-47\).
So the solutions are \((6,-2)\) and \((-9,-47)\).

Step 1: Set the equations equal

Set \(-\frac{1}{2}x + 2=x^{2}-6x + 8\).

Step 2: Multiply through by 2 to eliminate fraction

We get \(-x + 4 = 2x^{2}-12x + 16\).

Step 3: Rearrange into standard quadratic form

Rearrange to \(2x^{2}-11x + 12 = 0\).

Step 4: Use quadratic formula or factor (we'll use quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) where \(a = 2\), \(b=-11\), \(c = 12\))

First, calculate the discriminant \(\Delta=b^{2}-4ac=(-11)^{2}-4\times2\times12=121 - 96 = 25\).
Then \(x=\frac{11\pm\sqrt{25}}{4}=\frac{11\pm5}{4}\).
For the plus sign: \(x=\frac{11 + 5}{4}=\frac{16}{4}=4\).
For the minus sign: \(x=\frac{11-5}{4}=\frac{6}{4}=1.5\).

Step 5: Find the corresponding \(y\) values

For \(x = 4\), substitute into \(y=-\frac{1}{2}x + 2\): \(y=-\frac{1}{2}\times4+2=-2 + 2=0\).
For \(x = 1.5\), substitute into \(y=-\frac{1}{2}x + 2\): \(y=-\frac{1}{2}\times1.5+2=-0.75 + 2 = 1.25\).
So the solutions are \((1.5,1.25)\) and \((4,0)\).

The solution of a system of equations (in the context of graphing, especially for a line and a parabola) represents the point(s) where the graphs of the two equations intersect. The slope of a line is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\), the \(y\) - intercept is the value of \(y\) when \(x = 0\), and the vertex of a parabola is a specific point on the parabola (not just the solution of a system). So \(x = 4,y = 1\) represents the point where the line and parabola intersect.

Answer:

a. \((6, - 2)\) and \((-9, - 47)\)

Second Sub - Question (Solve the system \(y=-\frac{1}{2}x + 2\) and \(y=x^{2}-6x + 8\))