Problem 22 - Algebra 1

Question

Solve the equation. $$x+5=10$$

Step-by-Step Solution

Answer

The solution to the equation \(x + 5 = 10\) is \(x = 5\).

Step 1: Isolate the variable

Start by subtracting 5 from both sides of the equation. This will isolate x on one side. This gives: \(x + 5 - 5 = 10 - 5\).

Step 2: Simplify the equation

Now, simplify the equation by performing the subtraction. This simplifies to \(x = 5\).

Key Concepts

one-step equations

One-step equations are the most basic form of algebraic equations. They involve just one operation to solve.
These equations usually follow the basic structure of:

• adding a number to the variable,
• subtracting a number from the variable,
• multiplying a number with the variable, or
• dividing the variable by a number.

In our example, the one-step equation is given by: \(x + 5 = 10\). To find the value of \(x\), only one mathematical operation needs to be performed.
This simplicity makes them a good starting point for learning how to solve equations. Once you understand one-step equations, you can confidently move on to more complex problems, knowing the basic principles of solving equations are already in place.

isolation of variables

Isolation of variables means separating the variable to one side of the equation. This is essential to solving equations because it allows us to find the value of the unknown variable. In our example, the goal is to isolate \(x\) by getting it alone on one side of the equation.
We do this by performing an operation that will leave \(x\) by itself.

• Start with the equation, which is \(x + 5 = 10\).
• To isolate \(x\), we need to remove the \(+ 5\).
• Subtract 5 from both sides of the equation: \(x + 5 - 5 = 10 - 5\).
• The terms that are opposite (\(+5\) and \(-5\)) on the left side cancel out, leaving \(x\) on its own.

This leaves us with the simplified and solved equation, \(x = 5\).
By isolating the variable, we effectively determine what \(x\) must be to satisfy the original equation.

arithmetical operations

Arithmetical operations are the foundation of solving equations. They include addition, subtraction, multiplication, and division.
These operations are crucial tools that we use to manipulate equations and ultimately find the value of variables.
In our example, subtraction is the key operation used.

• The original equation is \(x + 5 = 10\).
• To solve it, we perform subtraction: \(x + 5 - 5 = 10 - 5\).
• We chose subtraction here because \(+5\) is present in the equation with \(x\).
• By subtracting 5 from both sides, we maintain the balance of the equation and solve it in one simple step.

When solving equations, choose the operation that best isolates the variable. This method allows us to simplify the equation and reach the correct solution.