$3$ . Which of the following accurately shows the first step when solving the following system of equations by substitution? $f(x)=-5 x^{2}+2 x+7$ $\newline$ $g(x)=-3 x+4$ $\newline$ $-5 x^{2}+2 x+7-63 x+4=0$ $\newline$ $g(x)=-3\left(-5 x^{2}+2 x+7\right)+4$ $\newline$ $-5 x^{2}+2 x+7=-3 x+4$ $\newline$ $f(x)=-5(-3 x+4)^{2}+2(-3 x+4)+7$

Full solution

3

. Which of the following accurately shows the first step when solving the following system of equations by substitution?

f(x)=-5 x^{2}+2 x+7

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g(x)=-3 x+4

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-5 x^{2}+2 x+7-63 x+4=0

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g(x)=-3\left(-5 x^{2}+2 x+7\right)+4

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-5 x^{2}+2 x+7=-3 x+4

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f(x)=-5(-3 x+4)^{2}+2(-3 x+4)+7

Identify Equations and Steps: Identify the equations given and the possible first steps for substitution. $\newline$ $f(x) = -5x^2 + 2x + 7$ $\newline$ $g(x) = -3x + 4$ $\newline$ Possible first steps: $\newline$ $1$ . $-5x^2 + 2x + 7 - 63x + 4 = 0$ $\newline$ $2$ . $g(x) = -3(-5x^2 + 2x + 7) + 4$ $\newline$ $3$ . $-5x^2 + 2x + 7 = -3x + 4$ $\newline$ $4$ . $f(x) = -5(-3x + 4)^2 + 2(-3x + 4) + 7$
Check Substitution Options: Check each option for logical substitution steps. $\newline$ Option $1$ : Incorrect, as it introduces an error in combining terms. $\newline$ Option $2$ : Incorrect, as it incorrectly substitutes and simplifies the expression. $\newline$ Option $3$ : Correct, sets one function equal to the other, a valid first step in substitution. $\newline$ Option $4$ : Incorrect, as it incorrectly substitutes and simplifies the expression.