Problem 62 - Algebra A Combined Function

Question

Solve for \(x\). $$ 2 x+3=5-2(3 x-1) $$

Step-by-Step Solution

Answer

x = \(\frac{1}{2}\)

Step 1: Distribute the 2

Begin by distributing the 2 on the right-hand side of the equation: \[ 5 - 2(3x - 1) = 5 - (6x - 2) = 5 - 6x + 2. \] This simplifies to: \[ 7 - 6x. \] So the equation becomes: \[ 2x + 3 = 7 - 6x. \]

Step 2: Move all x terms to one side

Add \(6x\) to both sides of the equation to move all the \(x\) terms to the left side: \[ 2x + 6x + 3 = 7. \] Simplify the expression: \[ 8x + 3 = 7. \]

Step 3: Isolate the term with x

Subtract 3 from both sides of the equation to isolate the term with \(x\): \[ 8x + 3 - 3 = 7 - 3. \] This simplifies to: \[ 8x = 4. \]

Step 4: Solve for x

Divide both sides by 8 to solve for \(x\): \[ x = \frac{4}{8}. \] Simplify the fraction: \[ x = \frac{1}{2}. \]

Key Concepts

Understanding the Distributive Property

The distributive property is a crucial mathematical rule that you will use often when dealing with equations. It shows how to multiply a single term across each term inside a parenthesis in an expression. In simpler words, it helps distribute multiplication over addition or subtraction.

Let's look at a common format: \( a(b + c) \). Here, the number or term \( a \) is outside the parenthesis and it gets distributed to both \( b \) and \( c \) within the parentheses.
Applying the distributive property gives us:

• \( a \times b \)
• \( a \times c \)

And together, these become: \( a \times b + a \times c \).
In the given exercise, you'd see the distributive property applied as \(-2(3x - 1)\), which turns into:

• \(-2 \times 3x = -6x\)
• \(-2 \times -1 = 2\)

This results in \(-6x + 2\) when combined. It changes the whole expression, giving us the ability to simplify further.
Using the distributive property correctly is key to transforming and solving complex problems.

Combining Like Terms

Once you've used the distributive property to eliminate any parentheses, the next essential step in simplifying an equation is combining like terms. Like terms are terms that have the same variable raised to the same power. This means they can be combined by simply adding or subtracting their coefficients.

Consider the equation segment: \( 2x + 6x + 3 = 7 \). Here:

• \( 2x \) and \( 6x \) are like terms because both have the variable \( x \) raised to the power of one.

You combine them by adding their coefficients:

• \( 2 + 6 = 8 \)

which results in \( 8x \), forming the simpler expression \( 8x + 3 \).

This makes the entire problem easier to solve, as it reduces the number of terms and focuses the equation. Remember, always look out for like terms; they make every equation more manageable.

Solving for a Variable

In many algebraic equations, the ultimate goal is to solve for a variable, often represented as \( x \). This means isolating \( x \) on one side of the equation, making it easier to see what it represents.

Consider the present equation: \( 8x + 3 = 7 \). The next step is to isolate \( x \) by moving all other numbers to the opposite side. Start by subtracting \( 3 \) from both sides:

• \( 8x + 3 - 3 = 7 - 3 \)
• \( 8x = 4 \)

This isolates the term containing \( x \). Now, you wish to make \( x \) alone, requiring division by the coefficient of \( x \), which is 8:

• \( x = \frac{4}{8} \)

After simplifying the fraction, you find \( x = \frac{1}{2} \).

This process, solving for a variable, is fundamental in algebra. It helps determine the value of unknown quantities, providing answers to various mathematical problems.