Select the correct answer from each drop-down menu. [tex]
\begin{array}{l}
y=x^2+2 x-1 \
y-3 x=5
\end{array}
[/tex] The pair of points representing the solution set of this system of equations is [ ]. and [ ].
Answer (2)
Solve the linear equation for y : y = 3 x + 5 .
Substitute the expression for y into the quadratic equation: 3 x + 5 = x 2 + 2 x − 1 .
Rearrange and factor the quadratic equation: x 2 − x − 6 = ( x − 3 ) ( x + 2 ) = 0 , which gives x = 3 or x = − 2 .
Find the corresponding y values: when x = 3 , y = 14 , and when x = − 2 , y = − 1 . Thus, the solution set is ( 3 , 14 ) and ( − 2 , − 1 ) .
Explanation
Problem Analysis We are given a system of two equations:
y = x 2 + 2 x − 1
y − 3 x = 5
We need to find the solution set (x, y) for the given system of equations.
Solving for y First, solve the second equation for y: y = 3 x + 5
Substitution Substitute this expression for y into the first equation: 3 x + 5 = x 2 + 2 x − 1
Rearranging the equation Rearrange the equation to form a quadratic equation in x: x 2 − x − 6 = 0
Factoring Factor the quadratic equation: ( x − 3 ) ( x + 2 ) = 0
Solving for x Solve for x: x = 3 or x = − 2
Finding y For each value of x, find the corresponding value of y using the equation y = 3 x + 5 .
If x = 3 , then y = 3 ( 3 ) + 5 = 14 .
If x = − 2 , then y = 3 ( − 2 ) + 5 = − 1 .
Solution Set The solution set is (3, 14) and (-2, -1).
Examples
Systems of equations are incredibly useful in real life. Imagine you're trying to figure out the break-even point for a new business. You have one equation for your costs and another for your revenue. Solving this system tells you exactly how much you need to sell to cover your expenses! This is just one example; systems of equations pop up in physics, engineering, economics, and many other fields.
The solution set for the system of equations is (3, 14) and (-2, -1). This was found by substituting y from the second equation into the first equation, solving the resulting quadratic equation, and finding the corresponding y values for each x. Therefore, the answer given is the pair of points that represent the solution set.
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