Problem 118 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 4 x+1+6 x=10 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{9}{10}\).

Step 1: Combine Like Terms

Begin by combining the similar terms on the left side of the equation. The terms \(4x\) and \(6x\) are like terms, so you can add them together:\[ 4x + 6x + 1 = 10 \]This simplifies to:\[ 10x + 1 = 10 \]

Step 2: Isolate the Variable Term

The next step is to isolate the \(10x\) term by removing the constant term \(+1\) from the left side of the equation. Do this by subtracting 1 from both sides of the equation:\[ 10x + 1 - 1 = 10 - 1 \]This simplifies to:\[ 10x = 9 \]

Step 3: Solve for the Variable

To solve for \(x\), divide both sides of the equation by 10:\[ x = \frac{9}{10} \]

Step 4: Conclusion

You have now isolated \(x\) and solved the equation. Therefore, the solution to the equation \(4x + 1 + 6x = 10\) is \(x = \frac{9}{10}\).

Key Concepts

Combining Like Terms

When working with linear equations, like terms are those terms within the equation that have the same variable raised to the same power. In the equation provided, we have two terms involving the variable \(x\): \(4x\) and \(6x\). Since these terms have the same variable, they can be combined by adding the coefficients together.

• Apply this concept by simply summing the coefficients of \(x\): \(4 + 6 = 10\).
• This process transforms the equation from \(4x + 1 + 6x = 10\) into \(10x + 1 = 10\).
• Combining like terms simplifies an equation, making it easier to solve for the unknown variable.

By recognizing and combining like terms, you reduce the complexity of an equation, setting a clear path toward finding the solution.

Isolating the Variable

Isolating the variable is a crucial step towards solving algebraic equations. The goal is to have the term with the variable by itself on one side of the equation. In our example, the equation after combining like terms becomes \(10x + 1 = 10\).

• Start by removing the constant term \(+1\) from the left-hand side to get the variable term by itself.
• To do this, perform the inverse operation of addition, which is subtraction.
• Subtract \(1\) from both sides, so: \(10x + 1 - 1 = 10 - 1\).
• This simplifies the equation to \(10x = 9\).

Removing constant terms simplifies the equation further and centers the problem around the variable, making the next steps more straightforward.

Solving for x

Once you have isolated the variable term \(10x\), the next task is to solve for \(x\). This means we want \(x\) left alone on one side of the equation. We currently have \(10x = 9\), so our goal is to manipulate this equation to become \(x = ?\).

• Here, \(10x\) implies 10 times \(x\). To undo this multiplication, use division.
• Divide every term of the equation by 10: \(\frac{10x}{10} = \frac{9}{10}\).
• This operation gives \(x = \frac{9}{10}\).

Solving for \(x\) finalizes the calculation, providing the value of the variable that satisfies the original equation. This result is the solution to the problem, where the value of \(x\) is completely determined.