Problem 12 - College Algebra Essentials

Question

Find all values of \(x\) satisfying the given conditions. \(y_{1}=10 x+6, y_{2}=12 x-7,\) and \(y_{1}\) exceeds \(y_{2}\) by 3.

Step-by-Step Solution

Answer

The value of \(x\) satisfying the given conditions is \(x = 2\).

Step 1: Substitute \(y_{1}\) and \(y_{2}\) into the equation

First, replace \(y_{1}\) with \(10x + 6\) and \(y_{2}\) with \(12x - 7\) in the equation \(y_{1} = y_{2} + 3\). This will give us an equation of the form \(10x + 6 = 12x - 7 + 3\).

Step 2: Simplify the equation

Simplify the right side of the equation by adding -7 and 3. This results in \(10x + 6 = 12x - 4 \).

Step 3: Move all terms involving \(x\) to one side

Subtract \(10x\) from both sides of the equation to get \(0 = 2x - 4\).

Step 4: Solve for \(x\)

To solve for \(x\), add 4 to both sides to get \(4 = 2x\), then divide both sides by 2. This will give \(x = 2\).

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can always be manipulated into the form \( ax + b = 0 \), where \( x \) represents the variable, and \( a \) and \( b \) are constants. The power of \( x \) is always 1.

For instance, the equations in the exercise, \( y_1 = 10x + 6 \) and \( y_2 = 12x - 7 \), are both linear equations. When we look to find a value of \( x \) where \( y_1 \) exceeds \( y_2 \) by 3, we effectively set up a new linear equation to solve the problem.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. For example, \( 10x + 6 \) and \( 12x - 7 \) are algebraic expressions where \( x \) represents the variable, and the numbers represent constants. Expressions become equations when there is an equality sign involved, which sets two expressions equal to each other.

In the given exercise, the expressions for \( y_1 \) and \( y_2 \) are equated to find the value of \( x \) that satisfies the condition. The expressions by themselves do not have a value unless applied within an equation or evaluated for a specific value of the variable.

Equation Simplification

Simplifying an equation means to reduce it to its most basic form, making it easier to solve. The simplification process often involves combining like terms, which are terms in the equation that have the same variable raised to the same power, or distributing multiplication over addition.

In the solution provided, simplification is demonstrated when \( 12x - 7 + 3 \) is simplified to \( 12x - 4 \). Simplifying both sides of the equation before attempting to isolate the variable makes the subsequent steps easier and more efficient.

Isolating Variables

Isolating the variable in an equation is the process of manipulating the equation to get the variable on one side by itself. This entails performing arithmetic operations such as addition, subtraction, multiplication, or division on both sides of the equation to maintain the equality.

In the context of our exercise, the variable \( x \) is isolated by first moving all terms involving \( x \) to one side, leading to \( 0 = 2x - 4 \). Then, the equation is further manipulated by performing addition and division until \( x \) is by itself on one side, resulting in the solution \( x = 2 \).