Problem 17 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 7-16, determine whether each value of \(x\) is a solution of the equation. Equation $$ 5 x-3=3 x+5 $$ Values (a) \(x=0\) (b) \(x=-5\) (c) \(x=4\) (d) \(x=10\)

Step-by-Step Solution

Answer

None of the values are a solution of the equation.

Step 1: Substitute \(x = 0\) into the equation

To check if \(x = 0\) is a solution, replace \(x\) with 0 in the equation. \[5(0) - 3 = 3(0) + 5\] After performing the multiplication and addition, check if the two sides of the equation are equal.

Step 2: Substitute \(x = -5\) into the equation

To check if \(x = -5\) is a solution, replace \(x\) with -5 in the equation. \[5(-5) - 3 = 3(-5) + 5\] After performing the multiplication and addition, check if the two sides of the equation are equal.

Step 3: Substitute \(x = 4\) into the equation

To check if \(x = 4\) is a solution, replace \(x\) with 4 in the equation. \[5(4) - 3 = 3(4) + 5\] After performing the multiplication and addition, check if the two sides of the equation are equal.

Step 4: Substitute \(x = 10\) into the equation

To check if \(x = 10\) is a solution, replace \(x\) with 10 in the equation. \[5(10) - 3 = 3(10) + 5\] After performing the multiplication and addition, check if the two sides of the equation are equal.

Key Concepts

Solution Verification

Solution verification is a key step in solving algebra equations. It helps ensure the accuracy of your answer.
Each solution to an equation must be validated to ensure that it satisfies the initial condition. When verifying a solution, substitute the suggested value back into the original equation. The goal is to check if both sides of the equation are equal after substitution. Here’s how it works:

• Substitute the solution value into the equation.
• Simplify each side of the equation.
• Compare the two sides to see if they are equivalent.

If both sides are equal, the solution is correct. For example, when checking if \( x = 0 \) is a solution for the equation \( 5x - 3 = 3x + 5 \), you substitute and simplify: \( 5(0) - 3 = 3(0) + 5 \) which simplifies to \( -3 = 5 \), and thus \( x = 0 \) is not a solution.

Linear Equations

Linear equations are mathematical statements of equality involving constants and variables.
They represent straight lines when graphed and are fundamental in algebra.
A linear equation is typically in the form of \( Ax + B = Cx + D \), where \( A \), \( B \), \( C \), and \( D \) are constants, and \( x \) is the variable. **Why linear?** Because the variable \( x \) is only to the first power.

• They have at most one solution.
• Their graph will always be a straight line.
• Constants affect the slope and position of the line.

Their simplicity makes them an excellent starting point for learning algebraic techniques. In our example, the equation is \( 5x - 3 = 3x + 5 \). It demonstrates a linear relationship because you can re-arrange terms to focus on finding \( x \). This equation structure prepares you to isolate the variable and find the god point at which both sides equate.

Substitution Method

The Substitution Method is a fundamental technique to solve equations, including systems of equations.
It's powerful for equations where direct simplification is challenging.
In substitution, replace the variable with a given number, derived from another equation or assumed value, to determine a solution. Here’s a quick guide:

• Choose the initial guess or instructions for substitution.
• Replace the term or variable in the equation with this guess.
• Simplify the equation to verify if both sides are equal.

For example, substituting \( x = 10 \) in the equation \( 5x - 3 = 3x + 5 \) involves replacing \( x \) with 10: \( 5(10) - 3 = 3(10) + 5 \) This simplifies to both sides equating to 47, confirming that \( x = 10 \) is indeed a solution. This method is especially useful in verifying potential solutions without solving the equation from scratch again.