Problem 46 - Prealgebra

Question

Find an equation that is equivalent to \(-9 t=18\)

Step-by-Step Solution

Answer

The equivalent equation is \(t = -2\).

Step 1: Analyze the Equation

The given equation is \(-9t = 18\). Our goal is to transform this equation into an equivalent form that is simpler.

Step 2: Isolate the Variable 't'

To isolate \(t\), divide both sides of the equation by \(-9\). This operation will help us solve for \(t\).

Step 3: Perform the Division

Divide both sides of the equation by \(-9\): \[ t = \frac{18}{-9} \]

Step 4: Simplify the Fraction

Simplify \(\frac{18}{-9}\) by dividing the numerator and the denominator by 9, which results in \(-2\). Therefore, \(t = -2\).

Step 5: Verify the Solution

Substitute \(t = -2\) back into the original equation to verify: \(-9(-2) = 18\), which simplifies to \(18 = 18\). This confirms our solution is correct.

Key Concepts

Equivalent Equations

Finding equations that are equivalent means transforming an original equation into another form that maintains the same solutions. It’s important because it allows us to work with forms that are simpler and easier to solve. This process does not change the solution set of the equation.

• For example, adding, subtracting, multiplying, or dividing both sides of an equation by the same number (except zero), results in an equivalent equation.
• In the exercise, \(-9t = 18\) was transformed to \(t = -2\) by dividing both sides by \(-9\).

This often helps simplify complex equations into something more manageable, keeping the relationship between variables intact.

Simplifying Fractions

Simplifying fractions is a key step in transforming equations into simpler forms. It involves reducing fractions to their simplest form where the numerator and the denominator have no common factors other than 1.

• In our example, the fraction \(\frac{18}{-9}\) was simplified by dividing the numerator and the denominator by their greatest common divisor, which is 9.
• This gives us \-2\, significantly simplifying the solution.

Recognizing when and how to simplify fractions is critical, as this makes both the computation and understanding of equations easier.

Isolation of Variables

Isolating a variable is a key technique in solving equations. It involves performing operations to get the unknown variable by itself on one side of the equation.

• The goal is to find the value of the unknown variable.
• For \(-9t = 18\), isolation was achieved by dividing both sides by the coefficient of \('t'\), which was \-9\.

This step inevitably leads to the solution of the equation, as it provides the variable’s value directly.

Verification of Solutions

Verification ensures the solution obtained truly satisfies the original equation. This is the process of checking your work to ensure accuracy.

• Plug the solution back into the original equation.
• If both sides of the equation are equal with this substitution, the solution is correct.
• In this case, substituting \(t = -2\) back confirms that \(-9(-2) = 18\). The equation holds true, verifying the accuracy of the solution.

Always verify solutions to confirm that no errors were made along the way.