Problem 43 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$4 x-7+2 x=9-10$$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Simplify the Left Side

Start by combining like terms on the left side of the equation. The terms with "x" are like terms: \[ 4x + 2x - 7 = (4+2)x - 7 = 6x - 7 \]

Step 2: Simplify the Right Side

The right side of the equation is a simple arithmetic problem: \[ 9 - 10 = -1 \]

Step 3: Write the Simplified Equation

Now, substitute the simplified left and right sides back into the equation, so it becomes: \[ 6x - 7 = -1 \]

Step 4: Isolate the Variable

To solve for \(x\), first add 7 to both sides to eliminate the constant term on the left: \[ 6x - 7 + 7 = -1 + 7 \] This simplifies to:\[ 6x = 6 \]

Step 5: Solve for x

Finally, divide both sides by 6 to solve for \(x\):\[ x = \frac{6}{6} \] This further simplifies to:\[ x = 1 \]

Key Concepts

Simplifying Equations

When dealing with linear equations, the first step is to simplify both sides of the equation as much as possible. This involves converting the equation into a simpler form, which makes it easier to handle later steps. To do this, you should:

• Eliminate any parentheses by distributing any constants across terms within the parentheses.
• Combine like terms, which we'll discuss more in the next section.

Remember that simplifying an equation doesn't change its equality; it just presents it in an easier-to-manage form. For example, if the equation on one side reads \( 4x + 2x - 7 \), simplify it by performing any arithmetic operations and combining similar terms. This approach helps manage the equation and brings clarity without altering its outcome.

Combining Like Terms

Combining like terms is an essential step in solving linear equations as it helps simplify the expression. 'Like terms' refer to terms that have identical variable parts. For example, in the expression \( 4x + 2x \), both terms are like as they contain the same variable raised to the same power.

Combining them involves adding or subtracting the coefficients while keeping the variable part unchanged. In our example, we do:

• Identify like terms: \(4x\) and \(2x\).
• Add the coefficients: \(4 + 2 = 6\).

This results in a simplified form: \(6x\). Such simplification reduces the complexity of the equation, paving the way for easier solving in subsequent steps.

Isolate the Variable

Once you've simplified both sides of the equation, the next strategy is to isolate the variable. This step is crucial because the ultimate goal of solving an equation is to find the value of the variable. To isolate the variable means getting it alone on one side of the equation.

Consider a simplified equation like \(6x - 7 = -1\). To isolate \(x\), you need to remove all other terms from the side it resides on using inverse operations. Start by:

• Adding 7 to both sides of the equation to cancel the -7. This yields \(6x = 6\).

This action leaves the variable term, \(6x\), isolated. Remember, whatever operation you perform on one side of the equation must be done to the other to maintain balance.

Dividing Both Sides

After isolating the variable term, the final step is to solve for the variable. In many linear equations, this involves dividing both sides by the coefficient of the variable.

For instance, after isolating \(6x = 6\), you should:

• Divide both sides by 6, the coefficient of \(x\).
• This yields: \(x = \frac{6}{6}\).

Simplifying this division shows that \(x = 1\). Dividing is crucial because it reduces the coefficient to 1, leaving the variable standing alone, providing a clear solution. Remember, dividing both sides is necessary to keep the equation balanced and correctly solved.