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Chapter 2: Problem 38

Solve each equation for the indicated variable. \(5 x-6 y=12\) for \(x\)

Short Answer

Expert verified
\(x = \frac{6y + 12}{5}\)

Step by step solution

01

Isolate the Term with 'x'

Start with the original equation: \[ 5x - 6y = 12 \] To solve for \(x\), we need to isolate the term containing \(x\). Add \(6y\) to both sides to remove it from the left side of the equation:\[ 5x = 6y + 12 \]
02

Solve for 'x'

Now that \(5x\) is isolated on one side of the equation, divide both sides by \(5\) to solve for \(x\):\[ x = \frac{6y + 12}{5} \] This simplifies \(x\) in terms of \(y\).

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Key Concepts

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

Solving for a variable
In algebra, solving for a variable means finding the value of an unknown in an equation. Variables are typically represented by letters such as \(x\), \(y\), or \(z\). In order to solve for a variable, one must isolate it on one side of the equation.To isolate a variable involves rearranging the equation using basic mathematical operations such as addition, subtraction, multiplication, and division. Basic Steps to Isolate a Variable:
  • Identify the term containing the variable you want to isolate.
  • Use inverse operations to move other terms away from the variable term. This essentially means doing the opposite operation. For instance, if something is added, you subtract it from both sides.
  • Once the variable is isolated, simplify the equation if necessary.
These operations do not change the equality; they only modify the layout to make isolating the variable possible. This is what we did in the example equation: adding \(6y\) to both sides, then dividing by \(5\). This technique is fundamental for various types of algebraic problems.
Linear equations
A linear equation is an equation that forms a straight line when graphed. The most straightforward form of a linear equation in two variables is \(ax + by = c\). In such an equation, \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.These equations are called 'linear' because they graph as straight lines. They hold the simplest relationship between variables, where the change in one variable results in a proportional change in another.Key Points:
  • Linear equations have one or more variables that are raised to the first power.
  • They often appear in the standard form \(ax + by = c\) or when \(y\) is isolated, in slope-intercept form: \(y = mx + b\).
  • Solving these equations involves simple arithmetic operations and often involves isolating a specific variable.
Understanding the structure of linear equations helps in recognizing the operations needed to solve for a variable, as illustrated in our example.
Step by step solutions
Step by step solutions in algebra provide a clear and logical path to solving equations. They help identify each action required to isolate variables and find their values.These solutions are especially helpful for those new to algebra as they clarify the problem-solving process, ensuring understanding rather than rote memorization.Benefits of Step by Step Solutions:
  • Encourages logical thinking by breaking down problems into manageable parts.
  • Helps track each operation, aiding in identifying mistakes.
  • Builds a strong foundational understanding of algebraic methods, facilitating the learning of more advanced topics.
By following a step by step method, such as isolating \(5x\) and then dividing by \(5\) in our example, learners gain confidence in their ability to solve equations systematically.

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Most popular questions from this chapter

For Problems \(1-16\), solve each equation. $$ |x+2|=9 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -2<3 x+4<2 $$

For Problems \(1-16\), solve each equation. $$ |5 x-7|=14 $$

Candace had scores of \(95,82,93\), and 84 on her first four exams of the semester. What score must she obtain on the fifth exam to have an average of 90 or better for the five exams?

Explain how you would solve the inequality \(|2 x+5|>-3\).
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