Problem 51 - Algebra 1: Concepts and Skills

Question

Rewrite the equation so that \(y\) is a function of \(x\). $$-8 x+2 y=10$$

Step-by-Step Solution

Answer

The equation, when rewritten with \(y\) as a function of \(x\), becomes \(y = 4x + 5\)

Step 1: Move the term with x to the other side of the equation

First, we'll move all terms involving \(x\) to the right side of the equation. We achieve this by adding \(8x\) to both sides of the equation. Doing so gives \(2y = 8x + 10\)

Step 2: Isolate y

Now we want to isolate \(y\) on one side of the equation. We can achieve this by dividing every term in the equation by 2. This simplifies the equation to \(y = 4x + 5\)

Key Concepts

Isolating Variables

Isolating variables is a fundamental technique in algebra that's essential for solving equations. It means rearranging the equation to make one variable the subject, which is typically on one side of the equation, with all other terms on the opposite side. The goal is to have this variable by itself, expressed as a function of the other variables.

To isolate a variable, we perform operations that 'undo' whatever is being done to it. For instance, if a variable is multiplied by a number, we can isolate it by dividing both sides of the equation by that number. Similarly, if a variable is being subtracted, we can isolate it by adding the same value to both sides.

Here are some tips for efficiently isolating variables:

• Identify the operations applied to the variable and apply inverse operations to both sides of the equation.
• Perform one operation at a time and simplify the equation at each step.
• Keep the variable terms on one side and the constants on the other to avoid confusion.

Using these techniques, students can simplify complex relationships and solve for one variable in terms of others, which is a foundational skill in algebra and beyond.

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. These equations form straight lines when graphed on a coordinate plane, and they are typically depicted in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

Key properties of linear equations include:

• They have one or two variables, typically \(x\) and \(y\).
• Each variable is to the first power (no exponents other than 1).
• The graph of a linear equation is a straight line.
• They often represent proportional relationships.

To solve a linear equation for a specific variable, like \(y\), we need to apply algebraic manipulation techniques to isolate \(y\). In the context of the original exercise \(\-8x + 2y = 10\), the solution demonstrates isolating \(y\) to express it as a function of \(x\), which is a fundamental aspect of understanding and working with linear relationships.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations. This kind of manipulation includes a variety of operations such as adding, subtracting, multiplying, and dividing both sides of the equation by a number, factoring, expanding expressions, and combining like terms.

Effective algebraic manipulation requires understanding these principles:

• Operations performed on one side of an equation must be done equally to the other side to maintain the balance.
• The distributive, associative, and commutative properties allow for the rearrangement and grouping of terms.
• To simplify expressions, combine like terms and reduce fractions where possible.

Some common steps in algebraic manipulation include moving terms to opposite sides of an equation, as in the exercise with \(\-8x + 2y = 10\), where adding \(8x\) to both sides leads to isolating \(y\). Mastery of algebraic manipulation is crucial for solving equations, understanding algebraic expressions, and it forms the basis for more advanced math concepts.