Question

Solve the equation. $$\frac{a}{-3}=7$$

Short answer

The solution to the equation \( \frac{a}{-3} = 7 \) is a = -21.

Step-by-step solution

Identify the equation

The equation to solve is \( \frac{a}{-3} = 7 \).

Isolate 'a'

To isolate 'a', multiply both sides of the equation by -3. This gives \( a = 7 \times -3 \).

Calculate 'a'

By performing the multiplication, 'a' equals -21.

Key concepts

Isolate Variable

Isolating the variable is a fundamental step in solving linear equations. The goal is to get the variable on one side of the equation alone. To achieve this when dealing with fractions, multiplication or division is often used to counteract the current operation affecting the variable.

For example, if a variable is divided by a number, as in the equation \(\frac{a}{-3} = 7\), you would multiply both sides of the equation by that number to isolate the variable. Here, multiplying both sides by -3 removes the fraction and achieves the goal: \( a = 7 \times -3 \). Always ensure to perform the same operation on both sides to maintain the equation's balance, which is a key rule in algebra.

Equation Operations

Equation operations are at the heart of solving mathematical problems. They involve adding, subtracting, multiplying, or dividing both sides of the equation by the same number to maintain equilibrium.

Imagine a set of scales in balance; whatever you do to one side, you must do to the other to keep the scales level. This is why when you have a variable that is affected by a negative number or fraction, you use the opposite or reciprocal operation to 'undo' that effect.

In our exercise, the operation is multiplication because the variable is initially divided. Algebraic operations should be done step by step while applying appropriate mathematical order of operations: first deal with parentheses, then exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.

Negative Numbers

Negative numbers in equations can confuse many students, but they are just numbers on the left side of zero on the number line. They play a crucial role in understanding the direction and magnitude of quantities.

When faced with a negative number in an equation, remember that multiplying or dividing by a negative flips the sign of the result. For instance, in our equation \(\frac{a}{-3} = 7\), when we isolate 'a' by multiplying both sides by -3, the negative sign of the -3 affects our solution. Thus, \(7 \times -3 = -21\), turning the positive 7 into a negative 21. This is an essential concept when dealing with real-world scenarios where quantities can increase or decrease, represented by positive and negative values respectively.