Problem 36 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Use multiplication to solve the equation. $$ -\frac{h}{3}=-16 $$

Step-by-Step Solution

Answer

The value of \(h\) that satisfies the initial equation is \(h=48\).

Step 1: Cancel out the negative sign

To cancel out the negative sign in front of the fraction, we multiply both sides of the equation by -1. The transformed equation is then \(\frac{h}{3}=16\).

Step 2: Cancel out the denominator

Since \(h\) is being divided by 3 in this equation, to cancel out the division, we multiply both sides of the equation by 3. This gives us \(h=16*3\).

Step 3: Calculate the value of h

Multiplying 16 by 3 we obtain \(h=48\).

Key Concepts

Equations with Fractions

When solving equations that include fractions, it's key to understand how to manipulate those fractions to make the problem simpler. In the given exercise, we encounter a negative fraction that needs to be solved for the variable h. To handle equations with fractions comfortably, here are useful steps to follow:

• Identify the fraction and the operation it is undergoing with the variable.
• If there's a negative sign, as with \( -\frac{h}{3}=-16 \), consider multiplying both sides of the equation by -1 to make the coefficient of the variable positive. This simplification, \(\frac{h}{3}=16\), is easier to visualize and solve.
• Next, to remove the fraction, we can multiply both sides by the denominator. Multiplying by the reciprocal of the fraction is another method which directly yields the variable without the fraction.
• Finally, simplify the equation to solve for the variable.

Grasping these steps ensures that equations involving fractions become less intimidating and more approachable.

Multiplying to Solve Equations

While solving equations, sometimes it’s necessary to multiply both sides of the equation by the same number to remove fractions or isolate the variable. This step keeps the equation balanced, meaning that the equality holds true. In the provided exercise, once the negative sign was handled, we had \(\frac{h}{3}=16\). Here, the variable h is divided by 3. To isolate h, we multiply both sides of the equation by 3, which is the denominator of the fraction. This operation essentially undoes the division by 3, following the rule that any number divided by itself equals 1 (except zero).

By multiplying both sides of the equation by 3, the equation simplifies to \(h = 16 \times 3\). Multiplication is a pivotal tool in solving such equations, and mastering this process allows for a straightforward path to finding the value of the variable.

Isolating Variables

Isolating the variable is a fundamental step in solving any algebraic equation. It means manipulating the equation so that the variable we're solving for is by itself on one side of the equal sign, and everything else is on the other side. In our exercise, after eliminating the negative sign, isolating the variable h was achieved by multiplying both sides by the denominator of the fraction. Once we had h alone on one side as in \( h = 48 \), we successfully isolated h and solved for its value.

Remember, whatever operation you perform, it must be done to both sides of the equation to maintain the balance. Be it multiplication, division, addition, or subtraction, maintaining the balance is key when isolating the variable. Mastery of this concept unlocks the ability to solve countless equations, as it is a technique commonly encountered in algebra.