Question

Solve the equation: \(3 x+4=8-x\).

Short answer

x = 1

Step-by-step solution

Isolate the variable on one side

Add x to both sides of the equation to get all the variable terms on one side:\[3x + 4 + x = 8 - x + x\]Which simplifies to:\[4x + 4 = 8\]

Eliminate the constant term from the variable side

Subtract 4 from both sides to isolate the variable term:\[4x + 4 - 4 = 8 - 4\]Which simplifies to:\[4x = 4\]

Solve for x

Divide both sides by 4 to solve for x:\[\frac{4x}{4} = \frac{4}{4}\]Which simplifies to:\[x = 1\]

Key concepts

Isolate the Variable

To solve linear equations effectively, the first step is to isolate the variable on one side of the equation. This essentially means we want all terms containing the unknown variable (in this case, x) to be on one side, and all constants on the other.

In our example:

• Start with the equation: 3x + 4 = 8 - x.
• Add x to both sides to move the variable term from the right-hand side to the left-hand side. By doing this, we get 3x + x + 4 = 8. It simplifies down to 4x + 4 = 8.

So, by adding x on both sides, we grouped all x terms together on the left side. This is our step towards isolating the variable.

Remember, whatever operation you perform on one side of the equation, you must do the exact same operation to the other side. This keeps the equation balanced.

Eliminate the Constant Term

The next key step is to eliminate the constant term from the side containing the variable. This helps in further isolating the variable.

Continuing with our example:

• We have the equation 4x + 4 = 8.
• Subtract 4 from both sides to get rid of the constant term on the left-hand side where our variable is. The operation looks like 4x + 4 - 4 = 8 - 4, which simplifies to 4x = 4.

This step is crucial because it ensures the variable term stands alone, making it clear and straightforward to solve for the variable in the next steps.

Just like before, whatever operation you perform on one side, do the same on the other side. This maintains the balance in the equation.

Divide to Solve for the Variable

Finally, we solve for the variable by dividing both sides of the equation by the coefficient of the variable.

In the equation we derived:

• We have 4x = 4.
• To solve for x, divide both sides by 4, resulting in x = 4/4, which simplifies to x = 1.

This method of dividing both sides by the coefficient ensures that the variable stands alone and provides the value directly.

In summary:

• We started with 3x + 4 = 8 - x.
• Reorganized it to 4x = 4 by isolating the variable and eliminating the constant term.
• Divided both sides by 4 to solve for x, giving us x = 1.

By understanding and applying these steps, isolating the variable, eliminating the constant term, and then dividing to solve for the variable, you can confidently solve linear equations.