Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 7 x+3=3 x-13 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -4\).

Step 1: Isolate terms involving \(x\)

The objective is to get all terms involving \(x\) on one side and constant terms on the opposite side of the equation. This is achieved by subtracting \(3x\) from both sides and subtracting \(3\) from both sides of the equation. The equation becomes \(7x -3x = -13 -3\).

Step 2: Simplify Both Sides

The left side simplifies to \(4x\) and the right side simplifies to \(-16\). Hence the equation is now \(4x = -16\).

Step 3: Solve for \(x\)

To isolate \(x\), divide both sides of the equation by \(4\). Thus, \(x = -16 / 4 = -4\).

Step 4: Check the Solution

Confirm the solution by substituting \(-4\) for \(x\) in the original equation. The equation \(7 * -4 + 3 = -28 + 3 = -25\) and \(3 * -4 - 13 = -12 - 13 = -25\). Since both sides of the equation equal each other, the solution is valid and the solution to the equation is \(x = -4\).

Key Concepts

Checking Solutions

Checking solutions is like verifying if the answer you got is correct. Whether you've plugged a value back into an equation or are confirming your work, this step ensures accuracy in solving linear equations.
When you check a solution, you are doing something like a mathematical detective work. You want to ensure the solution satisfies the original equation. It's important because arithmetic or algebraic mistakes can easily happen, and checking can help catch those slips.
For our exercise:

• First, find the original equation you solved, which is \(7x + 3 = 3x - 13\).
• Replace \(x\) with your calculated solution, \(-4\).
• Evaluate both sides of the equation to see if they are equal. This means you will do the math for each side of the equation.
• The left side becomes \(7(-4) + 3 = -28 + 3 = -25\).
• The right side becomes \(3(-4) - 13 = -12 - 13 = -25\).
• Since both evaluations equal \(-25\), your solution \(x = -4\) is correct.

Checking solutions helps you build confidence in your solution and makes sure you’ve done everything correctly.

Simplification of Equations

Simplification of equations is a crucial step in solving linear equations. It helps to reduce complex equations to simpler forms, making them easier to solve. The goal is to make the equation as simple as possible by combining like terms and simplifying both sides.
In the given exercise:

• Start by performing operations to get all terms with the variable \(x\) on one side. Here, you subtract \(3x\) from both sides of the equation \(7x + 3 - 3x = 3x - 13 - 3x\).
• This results in less clutter: \(4x + 3 = -13\). Now, subtract \(3\) from both sides to further simplify, leading to \(4x = -16\).
• Now the problem is much simpler, with \(x\) terms isolated on one side and constants on the other.

Simplification assists in making the equation easier to solve, effectively reducing it to a single step problem when done correctly.

Isolating Variables

Isolating variables means rearranging the equation so that the variable you are solving for is by itself on one side of the equation. This is often the heart of solving any algebraic equation.
To isolate the variable in a linear equation, you need to move all terms that do not contain the variable to the opposite side of the equation.
In the provided equation:

• After simplifying, you get \(4x = -16\).
• To isolate \(x\), divide both sides of the equation by \(4\) (the coefficient of \(x\)) to solve for \(x\).
• This results in \(x = -16 / 4\), which simplifies to \(x = -4\).
• Now, \(x\) is isolated and you have found the solution.

By isolating variables, you are zeroing in on the solution, making sure that it is expressed clearly and readied for validation through checking solutions.