Problem 70 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

Yes, 3 is a solution.

Step 1: Understand the Problem

We need to determine if the number \(3\) satisfies the equation \(x - 10 = -7\). This means substituting \(3\) for \(x\) and checking if the left side equals the right side of the equation.

Step 2: Substitute the Value

Substitute \(x = 3\) into the equation: \(3 - 10 = -7\).

Step 3: Simplify the Left Side

Calculate \(3 - 10\). This simplifies to \(-7\).

Step 4: Compare Both Sides

Now, compare the simplified left side, \(-7\), with the right side of the equation, which is also \(-7\). Since both sides are equal, the substitution satisfies the equation.

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving equations. It involves replacing a variable with a number to determine if it satisfies the equation. In our example, the equation given is \( x - 10 = -7 \). To use the substitution method, we substitute the proposed solution, which is the number 3, into the equation in place of \( x \).Here's how it works:

• Take the given value: here, it's \( x = 3 \).
• Replace \( x \) with 3 in the equation: Thus, the equation becomes \( 3 - 10 = -7 \).

By substituting the number 3 for \( x \), we can directly check if both sides of the equation are equal after simplification. This method is not only useful for checking solutions but also for finding unknowns in more complex algebraic problems.

Verifying Solutions

Verifying solutions is an essential step in solving equations to ensure that the substituted value satisfies the equation. After substituting \( x = 3 \) into the equation \( x - 10 = -7 \), we simplify the left side:

• Perform the operation \( 3 - 10 \), which gives \( -7 \).

Next, we compare it to the right side of the equation, which is also \( -7 \). Since both sides are equal, it confirms that the solution is correct.Verifying solutions can help you:

• Ensure no calculation errors were made.
• Confirm the correctness of a solution in algebraic problems.

This process is crucial, especially in complex equations, as it reassures that the solution makes sense within the given equation.

Linear Equations

Linear equations are equations that make a straight line when graphed on a coordinate plane. They often take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In our problem, the linear equation presented is \( x - 10 = -7 \).Characteristics of linear equations include:

• They have at most one variable with a degree of one.
• They produce a straight line graph.
• Solutions are found by isolating the variable using algebraic operations.

To solve, you might either manipulate the equation to isolate the variable or use methods like substitution to determine solutions. Understanding linear equations is the cornerstone of learning algebra as they form the basis for more complex equations and systems in mathematics.