Consider this system of equations:

[tex]\begin{array}{l}
y=4 x^2+2 x+9 \\
y-8=6 x
\end{array}[/tex]

Solved for [tex]y[/tex], the linear equation is $\square$
Which is the solution set of the system? $\square$

Solve the linear equation for y : y = 6 x + 8 .
Substitute the expression for y into the quadratic equation: 6 x + 8 = 4 x 2 + 2 x + 9 , which simplifies to 4 x 2 − 4 x + 1 = 0 .
Solve the quadratic equation: ( 2 x − 1 ) 2 = 0 , so x = 2 1 ​ .
Substitute x = 2 1 ​ back into the linear equation to find y : y = 6 ( 2 1 ​ ) + 8 = 11 . The solution set is {( 2 1 ​ , 11 )} ​ .

Explanation

Problem Analysis We are given the following system of equations:

{ y = 4 x 2 + 2 x + 9 y − 8 = 6 x ​
Our goal is to find the linear equation solved for y and the solution set of the system.

Solving the Linear Equation for y First, let's solve the linear equation for y . We have:

y − 8 = 6 x
Adding 8 to both sides, we get:
y = 6 x + 8

Substitution Now, substitute the expression for y from the linear equation into the quadratic equation:

6 x + 8 = 4 x 2 + 2 x + 9
Rearrange the equation to form a quadratic equation in standard form:
4 x 2 + 2 x + 9 − 6 x − 8 = 0
4 x 2 − 4 x + 1 = 0

Solving the Quadratic Equation Solve the quadratic equation 4 x 2 − 4 x + 1 = 0 . We can use the quadratic formula, x = 2 a − b ± b 2 − 4 a c ​ ​ , or we can notice that the quadratic is a perfect square:

( 2 x − 1 ) 2 = 0
So, 2 x − 1 = 0 , which gives x = 2 1 ​ .

Finding the Value of y Now, substitute the value of x back into the linear equation to find the corresponding value of y :

y = 6 x + 8
y = 6 ( 2 1 ​ ) + 8
y = 3 + 8
y = 11

Solution Set The solution set of the system is the ordered pair ( x , y ) = ( 2 1 ​ , 11 ) .

Final Answer The linear equation solved for y is y = 6 x + 8 , and the solution set of the system is {( 2 1 ​ , 11 )} .


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 level needed to cover all costs and start making a profit. This concept is crucial for financial planning and decision-making in business.

Answered by GinnyAnswer | 2025-07-03

The linear equation solved for y is y = 6x + 8, and the solution set of the system is {(1/2, 11)}.
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Answered by Anonymous | 2025-07-04