To solve the system of equations:
- \( y = -22x - 17 \)
- \( y = 3x^2 - 22x - 44 \)
we will set the two equations equal to each other since both are equal to \( y \):
\[ -22x - 17 = 3x^2 - 22x - 44 \]
Next, we can simplify this equation by adding \( 22x \) to both sides:
\[ -17 = 3x^2 - 44 \]
Now, we isolate the quadratic term:
\[ 3x^2 - 44 + 17 = 0 \] \[ 3x^2 - 27 = 0 \]
Next, we can add \( 27 \) to both sides:
\[ 3x^2 = 27 \]
Now, divide both sides by \( 3 \):
\[ x^2 = 9 \]
Taking the square root of both sides gives us:
\[ x = 3 \quad \text{or} \quad x = -3 \]
Now, let's find the corresponding \( y \) values for each \( x \):
- For \( x = 3 \):
\[ y = -22(3) - 17 = -66 - 17 = -83 \]
So, one solution is \( (3, -83) \).
- For \( x = -3 \):
\[ y = -22(-3) - 17 = 66 - 17 = 49 \]
So, the other solution is \( (-3, 49) \).
Putting it all together, the solutions to the system of equations are:
\[ (3, -83) \quad \text{and} \quad (-3, 49) \]