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Chapter 2: Problem 178

Solve the following equations with constants on both sides. $$2 a+8=-28$$

Short Answer

Expert verified
The solution is \(a = -18\).

Step by step solution

01

Identify the equation

The given equation is \( 2a + 8 = -28 \).
02

Isolate the variable term

Subtract 8 from both sides of the equation to isolate the term containing the variable \(a\): \[ 2a + 8 - 8 = -28 - 8 \]This simplifies to \[ 2a = -36 \].
03

Solve for the variable

Divide both sides of the equation by 2 to solve for \(a\): \[ \frac{2a}{2} = \frac{-36}{2} \]This simplifies to \[ a = -18 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the variable
When solving linear equations, the goal is to find the value of the variable. To do this, you'll often need to isolate the variable. Isolating the variable means manipulating the equation so that the variable is on one side and all the constants are on the other side. This process helps make the equation simpler and easier to solve. For the equation given,

**Step 1:** Start with the equation:
\(2a + 8 = -28\)

**Step 2:** Subtract 8 from both sides to eliminate the constant term on the left:
\(2a + 8 - 8 = -28 - 8\)
This isolates the variable term:
\(2a = -36\)

Running through each step ensures the variable term is isolated, making it easier to solve for the variable itself.
Linear equations
Linear equations are equations of the first degree, meaning their variables are not raised to any power other than one. They typically take the form \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants. In our example, the equation \(2a + 8 = -28\) is a linear equation. To solve linear equations:
  • Combine like terms if needed.
  • Isolate the variable using addition, subtraction, multiplication, or division.
  • Simplify the equation by performing the same operation on both sides.
Linear equations are foundational in algebra because they help us understand how quantities relate to each other.
Simplifying equations
Simplifying equations is an essential step in solving linear equations. It involves performing operations that make the equation easier to handle. These operations include:
  • Addition or subtraction to remove constants.
  • Division or multiplication to eliminate coefficients.
In our equation example:

**Step 1:** After isolating the variable term, we have:
\(2a = -36\)
**Step 2:** Divide both sides by 2 to solve for \(a\):
\(\frac{2a}{2} = \frac{-36}{2}\)
This simplifies to:
\(a = -18\)

Simplifying equations allows you to find the value of the variable accurately. Always remember to perform the same operation on both sides to maintain the balance of the equation.

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