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Chapter 1: Problem 53

Solve each of the following for \(a:(1,4)\) a. \(4 a-5=35\) b. \(\frac{3 a}{6}=-18\)

Short Answer

Expert verified
Part a: a = 10. Part b: a = -36.

Step by step solution

01

Part a: Isolate the variable

Start with the equation: \[4a - 5 = 35\]Add 5 to both sides to isolate the term with the variable:\[4a - 5 + 5 = 35 + 5\]Which simplifies to:\[4a = 40\]
02

Part a: Solve for the variable

Divide both sides by 4 to solve for \(a\):\[\frac{4a}{4} = \frac{40}{4}\]This simplifies to:\[a = 10\]
03

Part b: Simplify the equation

Start with the equation:\[\frac{3a}{6} = -18\]Simplify the fraction \(\frac{3a}{6}\) to \(\frac{a}{2}\):\[\frac{a}{2} = -18\]
04

Part b: Solve for the variable

Multiply both sides by 2 to solve for \(a\):\[2 \times \frac{a}{2} = -18 \times 2\]This simplifies to:\[a = -36\]

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Key Concepts

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

headline of the respective core concept
Isolating the variable is a crucial first step when solving equations. The goal is to get the variable by itself on one side of the equation. This involves undoing any operations that are being performed on the variable.
For example, in the equation \(4a - 5 = 35\), the variable \(a\) is being multiplied by 4 and then subtracted by 5. To isolate \(a\), we need to undo these operations in reverse order. First, we add 5 to both sides to cancel out the -5:
\(4a - 5 + 5 = 35 + 5\)
Which simplifies to:
\(4a = 40\)
By isolating the variable, we make it easier to move on to the next step鈥攕olving for the variable.
headline of the respective core concept
Simplification is the process of making an equation easier to solve. This can involve:
  • Combining like terms
  • Reducing fractions
  • Removing parentheses using the distributive property
Let's look at Part (b) of the exercise where we start with the equation: \(\frac{3a}{6} = -18\)
We need to simplify \(\frac{3a}{6}\). This can be reduced to \(\frac{a}{2}\) by dividing both the numerator and the denominator by 3:
\(\frac{a}{2} = -18\)
This simplified form makes it much easier to solve for the variable. Simplification helps to make complex equations more manageable.
headline of the respective core concept
Solving for the variable is the final step in solving an equation. Once we have isolated and simplified the equation, we can then find the value of the variable. For example, in Part (a), after isolating the variable, we have:
\(4a = 40\)
To solve for \(a\), divide both sides by 4:
\(\frac{4a}{4} = \frac{40}{4}\)
Which simplifies to:
\(a = 10\)
In Part (b), after simplifying, we have:
\(\frac{a}{2} = -18\)
To solve for \(a\), multiply both sides by 2:
\(2 \times \frac{a}{2} = -18 \times 2\)
Which results in:
\(a = -36\)
By following these steps, you can accurately determine the value of the variable in any equation.

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