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Chapter 8: Problem 49

Solve for \(z\) if \(x=1\) and \(y=-2\) $$ x+2 y+3 z=9 $$

Short Answer

Expert verified
The value of \(z\) is 4.

Step by step solution

01

Substitute Known Values

Substitute the known values for the variables into the equation: Given: - \(x = 1\) - \(y = -2\) Therefore, the equation becomes: \(1 + 2(-2) + 3z = 9\)
02

Simplify the Equation

Calculate the values of the substituted terms: - First, calculate \(2(-2)\): \(2(-2) = -4\) - Now substitute back: \(1 - 4 + 3z = 9\) - Simplify the left side: \(-3 + 3z = 9\)
03

Solve for \(z\)

To isolate \(z\), add 3 to both sides of the equation: \(-3 + 3z + 3 = 9 + 3 \) This simplifies to: \(3z = 12\) Finally, solve for \(z\) by dividing both sides by 3: \(z = \frac{12}{3} = 4\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
The substitution method is a useful approach for solving linear equations. It involves replacing the variable with its given value, making the problem simpler. In our exercise, we start by substituting the given values for the variables into the equation. The original equation is:
\( x + 2y + 3z = 9 \).

Given that \( x = 1 \) and \( y = -2 \), we substitute these values into the equation. This step reduces the equation to a simpler form. The equation now looks like this: \( 1 + 2(-2) + 3z = 9 \).

With substitution, we can focus on solving for the remaining variable, making the process straightforward.
simplification
Simplification is an essential step after substitution. It involves reducing the equation to make it easier to solve. After substituting the values, our equation becomes:
\( 1 + 2(-2) + 3z = 9 \).

Calculate the multiplication first: \( 2(-2) \) results in \( -4 \). Now the equation is:
\( 1 - 4 + 3z = 9 \).

Simplify by combining like terms on the left side: \( 1 - 4 \) equals \( -3 \), so we have:
\( -3 + 3z = 9 \).
By simplifying step-by-step, solving the equation becomes more manageable.
isolation of variables
Isolation of variables is key to solving equations. Once we simplify as much as possible, we isolate the target variable. In our simplified equation:
\( -3 + 3z = 9 \),

We need to get \( z \) on its own. First, add 3 to both sides to move the constant term:
\( -3 + 3z + 3 = 9 + 3 \),
which simplifies to:
\( 3z = 12 \).

Next, divide both sides by 3 to solve for \( z \):
\( z = \frac{12}{3} \), resulting in \( z = 4 \).

This method ensures you correctly solve for the variable by breaking down the steps systematically.

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Most popular questions from this chapter

Solve each system by the elimination method. $$ \begin{aligned} &\frac{1}{3} x+\frac{1}{2} y=\frac{13}{6}\\\ &\frac{1}{2} x-\frac{1}{4} y=-\frac{3}{4} \end{aligned} $$

An office supply store sells three models of computer desks: \(A, B,\) and \(C .\) In January, the store sold a total of 85 computer desks. The number of model \(B\) desks was five more than the number of model \(C\) desks, and the number of model \(A\) desks was four more than twice the number of model \(C\) desks. How many of each model did the store sell in January?

Solve each equation for \(y\). \(9 x-2 y=4\)

Solve each system by the elimination method. $$ \begin{array}{c} {5 x-2 y=3} \\ {10 x-4 y=5} \end{array} $$

Solve each system by the elimination method. $$ \begin{aligned} &7 x+2 y=0\\\ &4 y=-14 x \end{aligned} $$
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