Problem 29 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-\frac{1}{4} a+5=2$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 12\).

Step 1: Understand the Equation

The equation given is \(-\frac{1}{4}a + 5 = 2\). Our goal is to solve for \(a\), which is the unknown variable.

Step 2: Eliminate the Constant Term

To isolate the term with \(a\), subtract 5 from both sides of the equation. This step uses the addition property of equality.\[-\frac{1}{4}a + 5 - 5 = 2 - 5\]This simplifies to:\[-\frac{1}{4}a = -3\]

Step 3: Isolate the Variable

Now, we need to isolate \(a\). Multiply both sides of the equation by \(-4\) to get rid of the fraction:\[-4 \times \left(-\frac{1}{4}a\right) = -4 \times (-3)\]This simplifies to:\[a = 12\]

Key Concepts

Solving Equations

Solving an equation involves finding the value of the variable that makes the equation true. The given equation is \(-\frac{1}{4}a + 5 = 2\). Our aim is to determine the value of \(a\). The process generally includes manipulating the equation, using mathematical operations, to isolate the variable. These operations should keep the equation balanced, meaning whatever you do to one side should be done to the other as well. This is where properties like the addition property of equality come into play.

The addition property of equality states if you add (or subtract) the same value from both sides of an equation, the two sides remain equal. It helps in eliminating constant terms from one side to simplify the expression, setting the stage for isolating the variable. Once you've used this property to simplify the equation, you can more easily solve for the variable.

Isolation of Variables

Isolation of variables is a key part of solving equations, as it involves re-writing the equation such that the variable stands alone on one side. In our case, we are dealing with the equation \(-\frac{1}{4}a + 5 = 2\).

The first step to isolate \(a\) is to eliminate any constants connected to the variable. Here, the constant is 5. Subtract 5 from both sides of the equation using the addition property of equality, resulting in \(-\frac{1}{4}a = -3\).

• Subtracting identical values from both sides simplifies the equation.
• The goal is to have the variable term by itself on one side.

Next, to completely isolate \(a\), we remove the coefficient \(-\frac{1}{4}\) by performing an operation that enables us to leave the variable \(a\) by itself. This involves multiplying both sides by the reciprocal, \(-4\), because multiplying by a reciprocal cancels out the fraction. This results in \(a = 12\).

At this point, \(a\) is completely isolated, showing that when you plug \(a = 12\) back into the original equation, both sides equal.

Fraction Operations

Dealing with fractions is often intimidating, but grasping a few principles can ease the process. In the equation \(-\frac{1}{4}a + 5 = 2\), we have a fraction multiplying the variable \(a\).

To simplify, the first focus is getting rid of the fraction linked with the variable. In this case, we need to eliminate \(-\frac{1}{4}\) to make solving the equation easier. This is achieved by multiplying both sides by its reciprocal. Recall that the reciprocal is simply flipping the numerator and denominator of the fraction, so \(-\frac{1}{4}\) turns to \(-4\).

• Multiplying by the reciprocal "cancels" the fraction since \(-4 \times -\frac{1}{4} = 1\), leaving the variable itself.
• This keeps operations balanced and the equation solvable.

Understanding fraction operations like multiplication and finding reciprocals is crucial in equation solving, ensuring you can transition to a simpler form where variables are easily isolated and calculated.