Find the solution(s) of the system of equations:
[tex]
\begin{array}{l}
y=x^2+6 x+9 \
y=2 x+6
\end{array}
[/tex]
Answer (2)
Set the two equations equal to each other: x 2 + 6 x + 9 = 2 x + 6 .
Rearrange the equation to form a quadratic equation: x 2 + 4 x + 3 = 0 .
Solve the quadratic equation by factoring: ( x + 1 ) ( x + 3 ) = 0 , which gives x = − 1 and x = − 3 .
Substitute the values of x into y = 2 x + 6 to find the corresponding y values: y = 4 when x = − 1 , and y = 0 when x = − 3 . Thus, the solutions are ( − 3 , 0 ) and ( − 1 , 4 ) .
Explanation
Understanding the Problem We are given a system of two equations:
y = x 2 + 6 x + 9
and
y = 2 x + 6 .
We need to find the solution(s) of this system of equations, which are the points (x, y) that satisfy both equations.
Setting the Equations Equal To solve this system, we can set the two equations equal to each other:
x 2 + 6 x + 9 = 2 x + 6 .
Rearranging the Equation Now, let's rearrange the equation to form a quadratic equation in the standard form:
x 2 + 6 x + 9 − 2 x − 6 = 0 which simplifies to x 2 + 4 x + 3 = 0 .
Solving for x We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to 3 and add to 4. These numbers are 1 and 3. So, we can factor the equation as ( x + 1 ) ( x + 3 ) = 0 .
Therefore, the solutions for x are x = − 1 and x = − 3 .
Solving for y Now, for each value of x, we substitute it into either of the original equations to find the corresponding value of y. Using the linear equation y = 2 x + 6 is simpler.
If x = − 1 , then y = 2 ( − 1 ) + 6 = − 2 + 6 = 4 . So one solution is ( − 1 , 4 ) .
If x = − 3 , then y = 2 ( − 3 ) + 6 = − 6 + 6 = 0 . So the other solution is ( − 3 , 0 ) .
Final Answer Therefore, the solutions to the system of equations are ( − 3 , 0 ) and ( − 1 , 4 ) .
Examples
Systems of equations are used in various real-world applications. For instance, in economics, they can model supply and demand curves to find the equilibrium price and quantity. In physics, they can describe the motion of objects under certain constraints. In computer graphics, they are used to solve for the intersection of lines and surfaces. Understanding how to solve systems of equations is crucial for modeling and analyzing these types of problems.
The solutions to the system of equations are (-3, 0) and (-1, 4). To find these solutions, we set the equations equal to each other, rearranged them to form a quadratic equation, solved for x, and then substituted back to find corresponding y values.
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