Question

MULTIPLE CHOICE What is the first step you would use to solve \(\frac{1}{4}=-7 x ?\) (A) Divide by 4 (B) Multiply by 4. (C) Multiply by \(-7\) (D) Divide by \(-7\)

Short answer

The correct answer is (D) Divide by -7.

Step-by-step solution

Understanding the Problem

In any equation, the aim is to find the value of the unknown variable, in this case, 'x'. This equation is in the format ax = b, and the objective is to isolate 'x'. Inverse operations are used to cancel out operations, which makes them crucial in algebra when you're trying to isolate a variable.

Identify Inverse Operations

Here, 'x' is multiplied by -7. As the inverse operation of multiplication is division, the first step to isolate 'x' is to divide both sides of the equation by -7.

Selection of Option

By identifying the inverse operations needed, it can be determined that the correct option is (D) Divide by -7, as it's the process used to isolate 'x' in the given equation.

Key concepts

Inverse Operations

When solving linear equations, a fundamental concept is the use of inverse operations. Inverse operations are pairs of mathematical operations that cancel each other out, like addition and subtraction or multiplication and division. In the context of solving equations, they're the secret key to unlocking the values of variables.

For instance, if a variable is multiplied by a number, the inverse operation to release that variable is division by the same number. Similarly, if a number is added to the variable, it is isolated by subtracting the same amount. Knowing how to correctly apply these operations is essential in algebra, as it allows us to manipulate equations to our advantage and solve for unknown variables. These operations are quite literally the mirror image of the operation performed, reflecting and undoing each other to maintain the balance of the equation.

Isolate Variable

The primary goal in solving any algebraic equation is to isolate the variable—getting it alone on one side of the equation. Isolating the variable is like finding it in a busy street, clearing the crowd around it, and spotlighting it so it can shine unhindered. This often involves using inverse operations.

In the exercise given, the variable 'x' is trapped within the equation \(\frac{1}{4}= -7x\), surrounded by pesky numbers. To isolate 'x', these numbers need to be methodically moved to the other side of the equals sign, leaving 'x' in splendid isolation. This isolation process enables us to find the exact value of the variable, which is the end goal of solving linear equations. By mastering this art of isolation, students can navigate through more complex equations with confidence, always aiming to give their variable the solo performance it deserves.

Equation ax = b

The equation in the form \(ax = b\) is known as a linear equation because it represents a straight line when graphed on a coordinate plane. Here, 'a' and 'b' are constants, and 'x' is the variable. The solution to this equation will give the value of 'x' that makes the equation true.

To solve such an equation, you'll typically perform division as the last step. This means you divide the constant 'b' by the coefficient 'a' that is attached to 'x'. In the specific exercise we're looking at, the equation already looks different because it has a fraction. Remember that dividing by a fraction is equivalent to multiplying by its reciprocal. So, to get rid of the fraction \(\frac{1}{4}\) on one side, you'd usually multiply the entire equation by its reciprocal. However, since our equation has been multiply shifted (\frac{1}{4}= -7x\text{ instead of } ax = \frac{1}{4}\text{), the direct inverse operation is to divide by -7, completing the journey to isolate 'x' and solve for it.