Problem 84 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 3 a solution of \(x-10=-7 ?\)

Step-by-Step Solution

Answer

Yes, 3 is a solution because substituting it results in a true statement.

Step 1: Substitute the Given Number

To determine if 3 is a solution to the equation \(x - 10 = -7\), substitute \(3\) in place of \(x\) in the equation. The equation becomes \(3 - 10 = -7\).

Step 2: Perform the Arithmetic Operation

Now, perform the subtraction: \(3 - 10 = -7\). This simplifies to \(-7 = -7\).

Step 3: Compare the Results

Check if the result of the arithmetic operation, which is \(-7\), equals the right side of the original equation, which is also \(-7\). They're equal!

Key Concepts

Solution Verification

Verifying if a number is a solution to an equation is like solving a puzzle. You want to see if the number "fits" the equation. To verify, you substitute the number into the equation and check if both sides of the equation are equal. If they match, then the number is indeed a solution.

• Substitute the given number into the equation.
• Perform any needed calculations.
• Check if both sides of the equation equal the same value.

If everything checks out, the number is a solution; otherwise, it isn't. In our problem above, substituting 3 into the equation led to identical values on both sides of the equation, confirming that 3 is a valid solution.

Substitution Method

The substitution method is an essential technique in algebra for testing if a particular value makes an equation true. It involves replacing the variable in the equation with the given number.

• Take the given number and plug it into the equation wherever you see the variable.
• Ensure you perform the substitution correctly to avoid mistakes.

This method is straightforward but powerful, especially for simple equations. By using substitution, you directly test if the left-hand side of the equation equals the right-hand side when the variable is replaced by the number. In our case, substituting 3 for \( x \) in \( x - 10 = -7 \) leads to a simple arithmetic problem that can be quickly solved to verify the solution.

Arithmetic Operations

Arithmetic operations are the building blocks of solving equations. They include basic mathematical processes: addition, subtraction, multiplication, and division. In the example exercise, we mainly focus on subtraction.

• Identify the arithmetic operation in the equation.
• Perform the operation after the substitution.
• Check the results carefully to ensure accuracy.

In the equation \( 3 - 10 = -7 \), you subtract 10 from 3, which results in -7. This step is crucial because accurate arithmetic ensures that the left side and right side can indeed be compared fairly. Practicing these operations helps solidify the fundamentals of solving equations.

Linear Equations

Linear equations are the simplest type of equations you'll encounter in algebra. These are equations where the variable is not raised to any power higher than one. The general form of a linear equation is \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.

• Involve a variable such as \( x \) without exponents.
• Often easy to solve with one simple operation like addition or subtraction.
• Frequently seen in introductory algebra courses.

In the exercise, \( x - 10 = -7 \) is a linear equation. Solving such equations involves isolating the variable through basic arithmetic and checking whether a particular value, when substituted, satisfies the equation. Understanding linear equations is key as they form the foundation for more complex algebraic problem-solving.