Problem 75 - Prealgebra

Question

Solve each equation. $$85+x+24=180$$

Step-by-Step Solution

Answer

The solution is \( x = 71 \).

Step 1: Simplify the Left Side

First, simplify the left side of the equation by combining the constants. Add 85 and 24 together: \[ 85 + 24 = 109 \] So the equation becomes: \[ 109 + x = 180 \]

Step 2: Isolate the Variable

Next, subtract 109 from both sides of the equation to solve for \( x \):\[ 109 + x - 109 = 180 - 109 \]This simplifies to:\[ x = 71 \]

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, representing relationships between variables and constants with linear terms. A linear equation typically looks like this: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is a variable. Unlike quadratic or higher-degree equations, linear equations form straight lines when graphed on a coordinate plane.

The equation from the exercise, \( 85 + x + 24 = 180 \), is a classic example of a linear equation. Here, it's crucial that the degree of the variable, \( x \), is one, meaning that \( x \) is not raised to any power other than one.

This simplicity allows linear equations to have a consistent solution method. By learning how to solve linear equations, students build a solid mathematical foundation that is applicable to more complex problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or subtraction). In our equation, \( 85 + x + 24 \), each piece, \( 85 \), \( x \), and \( 24 \) are parts of the expression.

The goal in working with algebraic expressions is to simplify them by combining like terms. Like terms are terms that have the same variable raised to the same power. In our case, \( x \) is a variable term and \( 85 \) and \( 24 \) are constant terms. Since \( x \) stands alone without a coefficient here (which is implicitly 1), it doesn't combine with any other term in this expression.

Understanding how to identify and simplify parts of an algebraic expression is critical for efficient problem-solving in algebra. When expressions are as simple as possible, finding solutions becomes more straightforward.

Variable Isolation

Variable isolation is a crucial step in solving equations. It involves manipulating an equation to get the variable by itself on one side of the equation, making it easier to find its value.

In the example \( 109 + x = 180 \), isolating \( x \) means you have to eliminate any other numbers on the side of the equation with \( x \). This is done by performing operations that cancel out the constants. To solve for \( x \), you need to subtract 109 from both sides, simplifying the equation to \( x = 71 \).

Key strategies for variable isolation include:

• Perform the same operation on both sides to maintain balance.
• Focus on canceling out constants or coefficients attached to the variable.
• Check your work by plugging the solution back into the original equation to ensure it satisfies the equation.

By mastering variable isolation, solving any linear equation becomes a systematic process that leads directly to the solution.