Problem 73 - Algebra A Combined Function

Question

Solve each linear or quadratic equation \(4(x+6)+3=-3\)

Step-by-Step Solution

Answer

The solution is \( x = -\frac{15}{2} \).

Step 1: Distribute the 4

First, distribute the 4 to both terms inside the parenthesis. This will give you:\[ 4(x + 6) = 4x + 24 \] So, the equation becomes \[ 4x + 24 + 3 = -3 \].

Step 2: Combine Like Terms

Next, combine the like terms on the left side of the equation. Combine the 24 and 3 to get 27. This results in:\[ 4x + 27 = -3 \].

Step 3: Isolate the Variable Term

Subtract 27 from both sides of the equation to isolate the term with the variable on one side:\[ 4x = -3 - 27 \].This simplifies to:\[ 4x = -30 \].

Step 4: Solve for x

Finally, divide both sides of the equation by 4 to solve for \(x\):\[ x = \frac{-30}{4} \].Simplify the fraction:\[ x = -\frac{15}{2} \].

Key Concepts

Distributive Property

The distributive property is a vital concept in algebra. It helps us simplify expressions and solve equations. When you see an expression like \( a(b + c) \), apply the distributive property by multiplying \( a \) with both \( b \) and \( c \). This results in \( ab + ac \). In the given equation \( 4(x + 6) + 3 = -3 \), we first distribute the 4 inside the parenthesis:\[ 4 \, \times \, x + 4 \, \times \, 6 = 4x + 24 \]. This transforms the equation into \( 4x + 24 + 3 = -3 \). Remember, distributing terms ensures each component is correctly accounted for, paving the way for further simplification.

Combining Like Terms

Once the distributive property has been applied, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. They can be simplified by adding or subtracting their coefficients. In our equation \( 4x + 24 + 3 \), the like terms 24 and 3 are constants. Combining them results in 27. Thus, we can rewrite the equation as \( 4x + 27 = -3 \). Combining like terms is crucial as it simplifies the equation, making the solution process more straightforward.

Isolating the Variable

Isolating the variable is a key step to solving any equation. The goal is to get the variable on one side and the numbers on the other. Starting with \( 4x + 27 = -3 \), we want to isolate \( 4x \). Subtract 27 from both sides to achieve this:\[ 4x = -3 - 27 \]. Simplifying gives us \( 4x = -30 \). Isolating \( x \) in this way prepares us to find its specific value. This method is essential for finding solutions in linear equations.

Fractions in Equations

Fractions often appear in the final steps of solving an equation. Once the variable is isolated, as seen in \( 4x = -30 \), we divide both sides by 4 to solve for \( x \):\[ x = \frac{-30}{4} \]. This results in a fraction. Simplifying this fraction to its lowest terms gives \( x = -\frac{15}{2} \). When working with fractions, ensure they are expressed in simplest form for clarity and precision. Understanding fractions in equations is important as they often represent the final solution in real-world scenarios.