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Chapter 2: Problem 16

Solve each equation for \(y.\) $$ 7 x+8 y=11 $$

Short Answer

Expert verified
y = \frac{11 - 7x}{8}

Step by step solution

01

Isolate the term with y

Start by isolating the term with y on one side of the equation. Subtract 7x from both sides of the equation: \[ 8y = 11 - 7x \]
02

Solve for y

Divide both sides of the equation by 8 to solve for y: \[ y = \frac{11 - 7x}{8} \]

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Key Concepts

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

Isolation of variable
When solving linear equations, one of the most crucial steps is the isolation of the variable you want to solve for. In our example, we want to solve for the variable \( y \) in the equation: 7x + 8y = 11. Start by isolating the term with \( y \) on one side of the equation. This means we need to move any terms that do not include \( y \) to the other side. In our case, we subtract \( 7x \) from both sides to achieve this: 8y = 11 - 7x. Now, the term with \( y \) is isolated on the left side, while the other terms are on the right.
Division in equations
Once you have isolated the term with the variable, the next step is to solve for that variable. In our example, after isolation, we have the equation: 8y = 11 - 7x. To find the value of \( y \), we need to perform division. Divide both sides of the equation by 8: \( y = \frac{11 - 7x}{8} \). By dividing, we balance the equation and keep it equal. Using division helps us get our variable \( y \) by itself. Make sure to perform the division on both sides of the equation to maintain equality.
Linear equations in two variables
Linear equations in two variables involve equations where each term is either a constant or the product of a constant and a single variable. Commonly, these equations are written in the form \( ax + by = c \). In our example, the given linear equation is: 7x + 8y = 11. This equation has two variables, \( x \) and \( y \). To solve such equations, you may need to isolate one of the variables, as we did by isolating \( y \) and then solving for it: \( y = \frac{11 - 7x}{8} \). This solution represents \( y \) as a function of \( x \). These principles are fundamental when dealing with any linear equations in two variables.

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