Question
Solve the system of equations.
y=–14x–45
y=x2–3x–17
Write the coordinates in exact form. Simplify all fractions and radicals.
y=–14x–45
y=x2–3x–17
Write the coordinates in exact form. Simplify all fractions and radicals.
Answers
GPT-4o mini
To solve the system of equations
\[
y = -14x - 45
\]
and
\[
y = x^2 - 3x - 17,
\]
we will set the two equations equal to each other since they both equal \(y\):
\[
-14x - 45 = x^2 - 3x - 17.
\]
Next, we will rearrange the equation by moving all terms to one side:
\[
0 = x^2 - 3x - 17 + 14x + 45.
\]
Combining like terms gives:
\[
0 = x^2 + 11x + 28.
\]
Now, we can move all terms to the left side:
\[
x^2 + 11x + 28 = 0.
\]
Next, we will factor the quadratic equation. We look for two numbers that multiply to \(28\) and add to \(11\). The numbers \(4\) and \(7\) fit this requirement, so we can factor the equation as follows:
\[
(x + 4)(x + 7) = 0.
\]
Setting each factor to zero gives us the solutions for \(x\):
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4,
\]
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7.
\]
Now we will substitute these \(x\) values back into the first equation to find the corresponding \(y\) values.
1. For \(x = -4\):
\[
y = -14(-4) - 45 = 56 - 45 = 11.
\]
Thus, one solution is:
\[
(-4, 11).
\]
2. For \(x = -7\):
\[
y = -14(-7) - 45 = 98 - 45 = 53.
\]
Thus, another solution is:
\[
(-7, 53).
\]
Therefore, the solutions to the system of equations are:
\[
\boxed{(-4, 11)} \quad \text{and} \quad \boxed{(-7, 53)}.
\]
\[
y = -14x - 45
\]
and
\[
y = x^2 - 3x - 17,
\]
we will set the two equations equal to each other since they both equal \(y\):
\[
-14x - 45 = x^2 - 3x - 17.
\]
Next, we will rearrange the equation by moving all terms to one side:
\[
0 = x^2 - 3x - 17 + 14x + 45.
\]
Combining like terms gives:
\[
0 = x^2 + 11x + 28.
\]
Now, we can move all terms to the left side:
\[
x^2 + 11x + 28 = 0.
\]
Next, we will factor the quadratic equation. We look for two numbers that multiply to \(28\) and add to \(11\). The numbers \(4\) and \(7\) fit this requirement, so we can factor the equation as follows:
\[
(x + 4)(x + 7) = 0.
\]
Setting each factor to zero gives us the solutions for \(x\):
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4,
\]
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7.
\]
Now we will substitute these \(x\) values back into the first equation to find the corresponding \(y\) values.
1. For \(x = -4\):
\[
y = -14(-4) - 45 = 56 - 45 = 11.
\]
Thus, one solution is:
\[
(-4, 11).
\]
2. For \(x = -7\):
\[
y = -14(-7) - 45 = 98 - 45 = 53.
\]
Thus, another solution is:
\[
(-7, 53).
\]
Therefore, the solutions to the system of equations are:
\[
\boxed{(-4, 11)} \quad \text{and} \quad \boxed{(-7, 53)}.
\]