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Chapter 4: Problem 15

Solve the equations and inequalities. $$\frac{a}{2}+4=\frac{5}{2}$$

Short Answer

Expert verified
a = -3

Step by step solution

01

Isolate the variable term

Start by subtracting 4 from both sides of the equation to isolate the variable term. This gives us: \[ \frac{a}{2} + 4 - 4 = \frac{5}{2} - 4 \]Simplifying the right side, we get: \[ \frac{a}{2} = \frac{5}{2} - \frac{8}{2} \]thus,\[ \frac{a}{2} = -\frac{3}{2} \]
02

Solve for the variable

To solve for \(a\), multiply both sides of the equation by 2: \[ 2 \times \frac{a}{2} = 2 \times -\frac{3}{2} \]This simplifies to: \[ a = -3 \]

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Key Concepts

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

Isolating Variable
When solving linear equations, the first key step is often to isolate the variable. This means you separate the variable you are solving for from the other numbers in the equation.
Here's what it looks like:
In the problem \(\frac{a}{2}+4=\frac{5}{2}\), our goal is to isolate \(\frac{a}{2}\).
To do this, we start by eliminating the constant on the same side as the variable. Subtracting 4 from both sides helps us achieve this.
This leaves us with \(\frac{a}{2} + 4 - 4 = \frac{5}{2} - 4\), which simplifies to \(\frac{a}{2} = \frac{5}{2} - \frac{8}{2}\).
By only having the variable \(\frac{a}{2}\) left on one side, we have successfully isolated the variable.
Equation Simplification
Once we have isolated the variable, we need to simplify the equation.
Simplification involves performing arithmetic operations to combine like terms or to make the equation more manageable.
From our example, after isolating the variable, we have \(\frac{a}{2} = \frac{5}{2} - 4\).
To simplify, convert the constant on the right-hand side into a fraction: \(\frac{5}{2} - \frac{8}{2}\).
This gives \(\frac{a}{2} = \frac{5 - 8}{2} = \frac{-3}{2}\).
Using proper fractions and combining the numerators makes your equation clearer and simpler to solve.
Arithmetics in Algebra
Finally, to solve for the variable, you often use arithmetic operations鈥攁dding, subtracting, multiplying, or dividing鈥攖o clear any fractions or coefficients.
In our simplified equation, \(\frac{a}{2} = -\frac{3}{2}\), we need to clear the fraction to solve for \(a\).
Multiplication is the best tool here. Multiply both sides by 2 to eliminate the denominator:
\(\frac{a}{2} \times 2 = -\frac{3}{2} \times 2\), which simplifies to \(a = -3\).
By carefully applying arithmetic operations, you arrive at the solution seamlessly.
With practice, you become more comfortable and faster with these steps, making solving linear equations second nature.

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Most popular questions from this chapter

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{2 x-1}{5}-\frac{x-7}{3}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{x}{y}+\frac{y}{x}+\frac{3 x}{2 y}$$

Solve the equations and inequalities. $$\frac{w-2}{6}-2=\frac{w-1}{9}$$

Solve the equations and inequalities. $$\frac{w-2}{6}=\frac{w-1}{9}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A chemist wants to make \(90 \mathrm{ml}\) of a \(40 \%\) hydrochloric acid solution. She also wants to use all of the \(30 \mathrm{ml}\) of \(20 \%\) hydrochloric acid she has on hand. What percent solution must she use for the remaining 60 \(\mathrm{ml} ?\)
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