Solve the system of equations.
[tex]
\begin{array}{l}
y=x^2+13 x-49 \\
y=10 x-21
\end{array}
[/tex]
Answer (1)
Set the two equations equal to each other: x 2 + 13 x − 49 = 10 x − 21 .
Rearrange the equation to form a quadratic equation in x: x 2 + 3 x − 28 = 0 .
Solve the quadratic equation by factoring: ( x + 7 ) ( x − 4 ) = 0 , which gives x = − 7 and x = 4 .
Substitute the values of x into y = 10 x − 21 to find the corresponding y values: y = 19 when x = 4 and y = − 91 when x = − 7 . The solutions are ( 4 , 19 ) and ( − 7 , − 91 ) .
Explanation
Setting up the problem We are given a system of two equations:
y = x 2 + 13 x − 49
y = 10 x − 21
Our goal is to find the values of x and y that satisfy both equations. To do this, we can set the two expressions for y equal to each other.
Forming a quadratic equation Setting the two equations equal gives us:
x 2 + 13 x − 49 = 10 x − 21
Now, we rearrange the equation to form a quadratic equation in x :
x 2 + 13 x − 49 − ( 10 x − 21 ) = 0 x 2 + 13 x − 49 − 10 x + 21 = 0 x 2 + 3 x − 28 = 0
Solving for x Now we need to solve the quadratic equation x 2 + 3 x − 28 = 0 . We can try to factor it. We are looking for two numbers that multiply to -28 and add to 3. These numbers are 7 and -4.
So, we can factor the quadratic equation as:
( x + 7 ) ( x − 4 ) = 0
This gives us two possible solutions for x :
x + 7 = 0 ⇒ x = − 7 x − 4 = 0 ⇒ x = 4
Solving for y Now we substitute each value of x back into the linear equation y = 10 x − 21 to find the corresponding y values.
For x = 4 :
y = 10 ( 4 ) − 21 = 40 − 21 = 19
So, one solution is ( 4 , 19 ) .
For x = − 7 :
y = 10 ( − 7 ) − 21 = − 70 − 21 = − 91
So, the other solution is ( − 7 , − 91 ) .
Final Answer Therefore, the solutions to the system of equations are ( 4 , 19 ) and ( − 7 , − 91 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is represented by a quadratic equation and its revenue function is linear, solving the system of equations will give the production levels where the company breaks even. This helps in making informed business decisions about production and pricing strategies. The solution of the system, in this case, gives the points where the cost equals revenue, providing critical information for financial planning.
