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${:[8y=4x^(2)-12 x+46],[y=(3)/(2)x+(5)/(4)]:} If (a,b) is the solution to the system of equations shown, what is the value of a ?$

$\begin{array}{l} 8 y=4 x^{2}-12 x+46 \\ y=\frac{3}{2} x+\frac{5}{4} \end{array}$ $\newline$ If $(a, b)$ is the solution to the system of equations shown, what is the value of $a$ ?

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Unions and intersections of sets

Full solution

Q.

\begin{array}{l} 8 y=4 x^{2}-12 x+46 \\ y=\frac{3}{2} x+\frac{5}{4} \end{array}

\newline

If

(a, b)

is the solution to the system of equations shown, what is the value of

a

?

Identify Equations: Identify the system of equations to solve for the intersection point (a, b). $\newline$ The system of equations is: $\newline$ $8y = 4x^2 - 12x + 46$ $\newline$ $y = \frac{3}{2}x + \frac{5}{4}$
Substitute and Solve: Substitute the second equation into the first to solve for x. $\newline$ $8\left(\frac{3}{2}x + \frac{5}{4}\right) = 4x^2 - 12x + 46$
Simplify Equation: Simplify the equation by distributing $8$ on the left side. $\newline$ $12x + 10 = 4x^2 - 12x + 46$
Rearrange and Combine: Rearrange the equation to set it to zero and combine like terms. $\newline$ $4x^2 - 12x - 12x - 10 + 46 = 0$ $\newline$ $4x^2 - 24x + 36 = 0$
Divide and Simplify: Divide the entire equation by $4$ to simplify. $\newline$ $x^2 - 6x + 9 = 0$
Factor Quadratic: Factor the quadratic equation. $\newline$ $(x - 3)^2 = 0$
Solve for x: Solve for x by taking the square root of both sides. $\newline$ $x - 3 = 0$ $\newline$ $x = 3$
Conclude Solution: Now that we have the value of $x$ , which is $a$ , we can conclude the solution. $\newline$ The value of $a$ is $3$ .

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