Problem 2 - Elementary Algebra

Question

For Problems \(1-72\), use the properties of equality to help solve each equation. $$ x+7=21 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 14 \).

Step 1: Understanding the Equation

The equation given is \( x + 7 = 21 \). We need to solve for \( x \), which means finding the value of \( x \) that makes the equation true.

Step 2: Isolate the Variable

To solve for \( x \), we want to get \( x \) by itself on one side of the equation. To do this, we need to remove the 7 that is added to \( x \).

Step 3: Use the Subtraction Property of Equality

Subtract 7 from both sides of the equation to cancel out the +7 with \( x \). This gives us:\[ x + 7 - 7 = 21 - 7 \]

Step 4: Simplify the Equation

Simplifying both sides of the equation, we have:\[ x = 14 \]

Step 5: Verify the Solution

To ensure \( x = 14 \) is correct, substitute 14 back into the original equation:\( 14 + 7 = 21 \)Since both sides of the equation are equal, \( x = 14 \) is the correct solution.

Key Concepts

Properties of Equality

The properties of equality are fundamental principles in algebra that help us solve equations accurately. These properties maintain balance in an equation, ensuring that if we perform the same operation on both sides, the equality holds true.
One essential property is the addition property of equality, which states that you can add the same number to both sides of an equation without changing the equality.
Similarly, the subtraction property of equality, used in our exercise, allows us to subtract the same number from both sides, as done when solving the equation \( x + 7 = 21 \).

• Adding or subtracting the same number keeps the equation balanced.
• Multiplication and division properties work similarly, allowing for multiplying or dividing by the same nonzero number.

These properties are the building blocks for solving algebraic equations, ensuring each step maintains the equation's truth.

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable that satisfies the equation. The given equation, \( x + 7 = 21 \), is a simple linear equation, which means it can be visually represented as a straight line on a graph.
The key steps in solving such equations include isolation of the variable, a process in which operations are applied to both sides to leave the variable alone.
This often involves opposite arithmetic operations to cancel out terms. For example:

• In the equation \( x + 7 = 21 \), subtracting 7 from both sides isolates \( x \).
• Applying the reverse operation balances and solves the equation.

Linear equations typically have one solution, which can be found through systematic steps like those outlined in the solution process.

Step-by-Step Solutions

Step-by-step solutions in algebra provide a detailed pathway to solving equations. They are crucial in learning as they reveal the logical procession from a problem to its solution.
Breaking down problems into smaller tasks assists in understanding complex equations by focusing on one element at a time.
For the given equation, the solution is broken into steps:

• Step 1: Understand the problem - It begins with recognizing the equation \( x + 7 = 21 \).
• Step 2: Isolate the variable - Remove additional terms astride the variable.
• Step 3: Apply the subtraction property - Subtract 7 from both sides.
• Step 4: Simplify - After subtraction, the equation simplifies to \( x = 14 \).
• Step 5: Verify - Substitute back to ensure correctness.

This meticulous breakdown helps visual learners grasp complex algebraic concepts and builds a strong foundational understanding of solving equations.