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Chapter 3: Problem 27

Solve the equation using the multiplication or division properties of equality. $$-28=-7 t$$

Short Answer

Expert verified
The value of t is 4.

Step by step solution

01

Understand the Equation

The given equation is \(-28 = -7t\). Our goal is to solve for \( t \) by isolating it on one side of the equation.
02

Use the Division Property of Equality

To isolate \( t \), divide both sides of the equation by \( -7 \), the coefficient of \( t \).\( \frac{-28}{-7} = \frac{-7t}{-7} \)
03

Simplify Both Sides

Dividing yields \( 4 = t \). This means that \( t \) equals 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Division Property of Equality
The division property of equality is a crucial concept in algebra that helps us solve equations. This property tells us that if we divide both sides of an equation by the same non-zero number, the two sides remain equal.

Let's see it in action with our problem. The equation given is \-28 = -7t\. To isolate the variable \(t\), we need to get rid of the multiplication with -7. We use the division property of equality by dividing both sides of the equation by -7:
  • Left side: \( \frac{-28}{-7} \)
  • Right side: \( \frac{-7t}{-7} \)
This gives us \(4 = t\). By dividing both sides by -7, we've kept the equality balanced and isolated the variable \(t\). Remember, this only works if the number you are dividing by is non-zero.
Isolating Variables
Isolating the variable is the main goal when solving any equation. It means to get the variable we are looking for, \(t\) in our case, alone on one side of the equation.

In the problem \-28 = -7t\, the variable \(t\) is multiplied by -7. To isolate \(t\), we need to perform the opposite operation 鈥� division. By dividing both sides of the equation by -7, as shown in the previous section, the -7 cancels out on the right side:
  • Left side becomes \( \frac{-28}{-7} \)
  • Right side becomes \( \frac{-7t}{-7} \)
The result is \(t = 4\). Now, the variable \(t\) is isolated and we have solved for it. This process of doing the opposite operation is key to isolating variables.
Simplifying Equations
Simplifying equations is an important part of solving them. Simplification makes equations easier to understand and solve.

After applying the division property of equality to our equation \-28 = -7t\, we simplified it to \(4 = t\). What does it mean to simplify?
  • It involves reducing the equation to its simplest form.
  • Getting an equation where the variable is isolated on one side.
  • Ensuring both sides of the equation are as straightforward as possible.
In this case, simplifying involved dividing both sides by -7, which reduced our equation to a straightforward form: \(4 = t\). This is a simple equation that tells us the value of \(t\) directly.

Simplifying equations often involves combining like terms, reducing fractions, or performing operations that clear parentheses, but the goal is always to make the equation as easy to interpret as possible.

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Most popular questions from this chapter

Apply the distributive property. $$-4\left(3-n^{2}\right)$$

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Apply the appropriate property to simplify the expression. $$-8(4-p)$$

Solve the equation. $$-10-4=6 m-5(3+m)$$

Comparing the Properties of Equality In your own words, explain how to determine when to use the subtraction property of equality.
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