Problem 22 Solve. $$3 x-7 y=2, \text { fo... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 1: Problem 22

Solve. $$3 x-7 y=2, \text { for } y$$

Short Answer

Expert verified

The solution for $ y $ is $ y = \frac{3x - 2}{7} $.

Step by step solution

Isolate the term involving y

Start with the given equation: \[ 3x - 7y = 2 \]. To isolate the term involving $ y $, subtract $ 3x $ from both sides: \[ -7y = 2 - 3x \].

Solve for y

To solve for $ y $, divide both sides of the equation by $ -7 $: \[ y = \frac{2 - 3x}{-7} \].

Simplify the expression

Simplify the fraction: \[ y = \frac{3x - 2}{7} \]. Now, the equation is solved for $ y $.

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Key Concepts

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

isolation of variable

To solve a linear equation for a specific variable, the first crucial step is to isolate that variable. This means you want to get the variable by itself on one side of the equation. In our exercise, we aim to solve for y in the equation \[ 3x - 7y = 2 \]. To do this, we need to isolate the term that contains y. We start by subtracting 3x from both sides of the equation, which gives us: \[ -7y = 2 - 3x \]. This operation moves all terms involving x to the right-hand side and leaves the term with y on the left-hand side, making it easier to solve for y in the next steps.

manipulating algebraic expressions

Manipulating algebraic expressions is a fundamental skill in solving equations. It involves performing the same operation on both sides to maintain equality. In our exercise, after isolating the y-term (\[ -7y = 2 - 3x \]), we continue by isolating y completely. To do this, we divide both sides by -7: \[ y = \frac{2 - 3x}{-7} \]. This division ensures that y is by itself. Remember that when you divide or multiply both sides of an equation by the same non-zero number, you maintain the equality. This manipulation is crucial to solving for y.

simplifying fractions

Simplifying fractions is the final step in our exercise. Simplification makes an expression cleaner and easier to understand. After manipulating the algebraic expression to \[ y = \frac{2 - 3x}{-7} \], we notice that both the numerator and the denominator can be rewritten for simplicity. We rewrite the expression as \[ y = \frac{3x - 2}{7} \]. Here's why: negative signs in the numerator and denominator cancel out if they both change signs. So \[ \frac{2 - 3x}{-7} \] is equivalent to \[ \frac{-(2 - 3x)}{-7} \], then further simplified to \[ \frac{3x - 2}{7} \]. By simplifying, we present the solved equation for y in its cleanest form.

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91影视

Solve. $$3 x-7 y=2, \text { for } y$$

Short Answer

Step by step solution

Isolate the term involving y

Solve for y

Simplify the expression

Key Concepts

isolation of variable

manipulating algebraic expressions

simplifying fractions

One App. One Place for Learning.

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