Problem 78 Solve for the specified variable... [FREE SOLUTION]

Chapter 1: Problem 78

Solve for the specified variable. (See Examples $8-9)$ $$ 7 x-2 y=5 \text { for } y $$

Short Answer

Expert verified

y = \frac{7x}{2} - \frac{5}{2}

Step by step solution

Move the constant term

To solve for y, first isolate the term involving y. Subtract $7x$ from both sides of the equation: \[ 7x - 2y - 7x = 5 - 7x \] This simplifies to: \[ -2y = -7x + 5 \]

Solve for y

Next, divide each term by -2 to isolate y: \[ y = \frac{-7x + 5}{-2} \] This can be simplified further by dividing each term in the numerator by -2: \[ y = \frac{-7x}{-2} + \frac{5}{-2} \]\[ y = \frac{7x}{2} - \frac{5}{2} \]

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Key Concepts

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

Variable Isolation

Understanding how to isolate a variable is a critical skill in solving linear equations. This process involves manipulating the equation so that the desired variable is by itself on one side of the equation.

For example, in the equation \[ 7x - 2y = 5 \], we were asked to solve for y. To start, you need to move all terms not involving y to the other side of the equation.

Here's how:

First, we have 7x and -2y. To isolate the term involving y, subtract 7x from both sides of the equation: \[ 7x - 2y - 7x = 5 - 7x \]
The equation then simplifies to \[-2y = -7x + 5 \].

With y now isolated in terms of x, we can solve for y by further manipulation.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. They take the form of \[ ax + by = c \], where a, b, and c are constants.

In the given problem, we start with the linear equation \[7x - 2y = 5\]. The process of solving such equations often involves isolating the variable of interest (in this case, y).

Remember, to keep the equation balanced, any operation performed on one side must also be done to the other side. This idea is essential in maintaining the equation's equality while solving it.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations using algebraic rules. It's essential for solving equations, like the given linear equation where we solved for y.

After isolating the term involving y, we had the equation \[-2y = -7x + 5 \]. The next step was to isolate y completely by dividing each term by -2:

\[ y = \frac{-7x + 5}{-2} \]
This further simplifies by dividing each term in the numerator by -2:
\[ y = \frac{-7x}{-2} + \frac{5}{-2} \]
Which simplifies to \[ y = \frac{7x}{2} - \frac{5}{2} \]

Practicing algebraic manipulation will make solving linear equations more intuitive.

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91影视

Solve for the specified variable. (See Examples \(8-9)\) $$ 7 x-2 y=5 \text { for } y $$

Short Answer

Step by step solution

Move the constant term

Solve for y

Key Concepts

Variable Isolation

Linear Equations

Algebraic Manipulation

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