Solve for y in the equation y = x^2 + 3x - 10, substitute x = 7, and solve the system of equations given in Example 1.27.

Understand the Problem

The question is asking for a solution involving equations and potentially a system of linear equations. It features operations on a quadratic function and appears to pertain to a specific example in a mathematical context.

Answer

The value of $y$ is $60$.

Steps to Solve

Evaluate the Quadratic Function at $x = 7$

Substituting $x = 7$ into the quadratic equation $y = x^2 + 3x - 10$:

$$ y = 7^2 + 3(7) - 10 $$

Calculating $7^2 = 49$, $3(7) = 21$, so:

$$ y = 49 + 21 - 10 $$

Simplify the Equation

Now, combine the terms:

$$ y = 49 + 21 - 10 = 60 $$

Thus, when $x = 7$, the value of $y = 60$.

Set Up the System of Equations

For the second part, we analyze the system given:

$$ \begin{align*}

& \quad 4x + 3y + b_2 = 25 \
& \quad x + 5y + 7z = 13 \
& \quad 2x + 9y = 1 \end{align*} $$
Express in Matrix Form

The system can be expressed in augmented matrix form as follows:

$$ \begin{bmatrix} 4 & 3 & 6 & | & 25 \ 1 & 5 & 7 & | & 13 \ 2 & 9 & 1 & | & 1 \end{bmatrix} $$

Solve the System

Apply methods such as substitution or elimination to solve for the variables $x$, $y$, and $z$ in the equations structured above, which will allow determining the values that satisfy all equations.

The value of $y$ when $x = 7$ is $60$.

More Information

This solution details the evaluation of a quadratic function at a specific point and provides a framework for solving a system of linear equations, often used in algebra.

Tips

Forgetting to correctly apply the order of operations when evaluating the function. Always calculate powers and multiplication before addition and subtraction.
Misinterpreting the setup of the system of equations and neglecting to keep the equations balanced when manipulating them.

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