Problem 6 - College Algebra Essentials

Question

Solve and check linear equation. \(3 x+5=2 x+13\)

Step-by-Step Solution

Answer

The solution for the linear equation \(3x+ 5 = 2x + 13\) is \(x = 8\).

Step 1 - Isolate variable terms on one side

Subtract \(2x\) from both sides of the equation. This gives \(3x - 2x = 13 - 5\), which simplifies to \(x = 13 - 5\)

Step 2 - Finding the Variable

Now do the subtraction on the right-hand side. This gives \(x = 8\)

Step 3 - Checking Solution

Substitute x=8 into the original equation \(3 x+5=2 x+13\), to check solution, if left hand side equals right hand side, the solution is valid.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value of a variable that makes an equation true. The process is like solving a puzzle, where each step brings you closer to the solution. You must perform the same operation on both sides of an equation to keep it balanced. Think of it as a seesaw, where both sides need to be equal for it to be in balance.

Steps to solving an equation usually include:

• Isolating the variable term to one side of the equation.
• Simplifying both sides of the equation if necessary.
• Solving the simplified equation for the unknown variable.
• Checking your solution by substituting back into the original equation.

When you subtract, add, multiply, or divide, remember to do so to both sides of the equation! This ensures that the equation stays valid throughout the process.

Algebraic Equations

Algebraic equations are mathematical statements indicating that two expressions are equal. These equations can range from simple equations with one variable, like the one we are solving, to more complex systems involving several variables. The particular equation we are considering is a linear equation, which means it forms a straight line when graphed.

An essential characteristic of linear equations is that they contain no exponents higher than one, such as in our example: \[3x + 5 = 2x + 13.\]

Understanding how to manipulate these equations is crucial. Here are some key points to remember:

• Linear equations have one solution when they involve one variable.
• They form the backbone of algebra and are foundational for understanding more intricate algebraic concepts.
• They always graph as straight lines on a coordinate plane, which can make visualization easier.

Mathematical Solutions

In mathematics, finding a solution refers to determining the value(s) of variable(s) that satisfy the equation. The solution provides the answer to the posed problem and confirms that the left and right sides of the equation are indeed equal.

For the equation \[3x + 5 = 2x + 13,\]
the solution can be found by first simplifying the terms and isolating the variable. Once \(x = 8\) is obtained, it is important to verify this solution. Verification reassures that no mistake was made during calculation.

Here’s how you check your solution:

• Substitute the value back into the original equation.
• Simplify both sides of the equation to see if they balance.
• If both sides are equal, your solution is correct!

Always double-checking helps deepen understanding and builds confidence for tackling more challenging equations.