Problem 59 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(y=10 x+16\) for \(x\).

Step-by-Step Solution

Answer

Question: Given the equation \(y=10x+16\), solve for variable \(x\) and express it in terms of \(y\). Answer: \(x = \frac{y-16}{10}\)

Step 1: Identify the given equation

The given equation is \(y=10x+16\) and we need to solve for variable \(x\).

Step 2: Isolate x term on one side

To isolate \(x\), we can first subtract 16 from both sides of the equation. This gives us: \(y-16 = 10x\)

Step 3: Solve for x

Now, we need to express x in terms of y. To do this, we will divide both sides of the equation by 10: \(\frac{y-16}{10} = x\) Now we have the equation solved for x, where x is expressed in terms of y: \(x = \frac{y-16}{10}\)

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Think of them as sentences in the language of mathematics, where variables represent the unknown elements we are interested in solving for. For instance, in the expression \(y = 10x + 16\), \(y\) and \(x\) are variables, 10 and 16 are constants, and the equals sign \(=\) and the addition sign \(+\) are the operation symbols connecting them.

When working with algebraic expressions, it's essential to understand the properties of equality and operations as they allow us to manipulate these expressions effectively. This understanding enables us to reconfigure the expressions to solve for a particular variable, as needed in various algebra problems. It is also crucial to learn how to simplify these expressions by combining like terms or using the distributive property, which will often be the first step in solving many algebraic problems.

• Constants are like fixed markers that help us measure how far or with what intensity the variables are related.
• Variables represent unknown quantities and can change, giving algebra its flexibility to solve for them.
• Operation symbols guide us in combining these elements following mathematical rules.

Isolating Variables

Isolating a variable is a technique used to solve an equation for one particular variable. This often means getting the variable alone on one side of the equation, while the rest of the terms are on the other side. In our exercise, we have the equation \(y = 10x + 16\), and we are tasked with isolating \(x\).

To accomplish this, we first remove the constant term from the side with the variable by performing the inverse operation. Since 16 is added to \(10x\), we subtract 16 from both sides of the equation, maintaining the balance of the equation:
\[y - 16 = 10x\]. Next, since \(x\) is multiplied by 10, we perform the inverse operation by dividing both sides by 10, resulting in \(x\) being isolated:
\[x = \frac{y - 16}{10}\].

This step is crucial in solving not just linear equations but virtually any equation where a single solution for a variable is required.

• Start by moving terms that do not contain the variable of interest to the other side using addition or subtraction.
• Then, use multiplication or division to rid the coefficient attached to the variable.

Manipulating Equations

Manipulating equations involves rearranging and transforming equations using mathematical operations to achieve a specific form or solve for a variable. During this process, it's vital to maintain the equation's balance, meaning whatever you do to one side must be done to the other. This is based on the properties of equality.

In the example given, the equation \(y = 10x + 16\) was manipulated by first subtracting 16 from both sides, then dividing both sides by 10 to isolate \(x\). The goal here was to manipulate the original equation into a form where \(x\) stands alone, thus 'solving for \(x\).'

Key principles of equation manipulation include:

• Performing the same operation on both sides so as not to disrupt the balance of the equation.
• Using the inverse operations – addition/subtraction and multiplication/division – to simplify or isolate terms.
• Understanding that sometimes, more complex operations like factoring, taking square roots, or applying trigonometric identities might be necessary.

Whatever the operation, the goal of manipulating equations is always to maintain equality while transforming the equation's structure to find a solution.