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Chapter 3: Problem 42

Use the properties of equality to help solve each equation. $$ \frac{n}{-9}=-5 $$

Short Answer

Expert verified
The value of \( n \) is 45.

Step by step solution

01

Identify the Equation

The given equation is \( \frac{n}{-9} = -5 \). This indicates that the variable \( n \) divided by \(-9\) results in \(-5\).
02

Use Multiplication Property of Equality

To solve for \( n \), multiply both sides of the equation by \(-9\) to eliminate the division by \(-9\). This gives us:\[ \left( \frac{n}{-9} \right) \times (-9) = (-5) \times (-9) \]
03

Simplify Both Sides

On the left side, multiplying \( \frac{n}{-9} \) by \(-9\) cancels out the \(-9\), leaving \( n \). On the right side, \(-5 \times -9 = 45\). So the equation simplifies to:\[ n = 45 \]

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Key Concepts

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

Solving Equations
Solving equations is like solving a puzzle. We look for the unknown piece that makes everything fit perfectly. In math, this unknown is often represented by a variable, such as **\( n \)**. The goal is to find the value of this variable that makes the equation true.
To solve an equation, we perform operations to simplify and figure out what the variable equals. These operations must keep both sides of the equation balanced, much like keeping a scale level.
  • Identify the variable and its operation in the equation.
  • Use mathematical operations to isolate the variable on one side.
  • Whatever operation you perform on one side, you must also perform on the other.
Multiplication Property of Equality
The multiplication property of equality is a handy tool when solving equations. It states that if you multiply both sides of an equation by the same number, the equation remains balanced.
This property is particularly useful when you need to eliminate a fraction or a divisor from the equation. For instance, if the equation is \( \frac{n}{-9} = -5 \), multiplying both sides by \(-9\) helps remove the division by \(-9\).
  • Determine the number you need to multiply by to cancel out fractions or divisors.
  • Apply the multiplication consistently to both sides of the equation.
  • Rebalance the equation if necessary by simplifying any resulting expressions.
By using multiplication, we can make the variable stand alone, which is a crucial step in finding its value.
Simplification of Equations
Simplification of equations involves reducing them to their most basic form, making them easier to solve.
Once you've applied the multiplication property of equality, you often need to simplify the equation further. This may involve simplifying terms on either side and ensuring the variable is isolated.
In the case of \( \frac{n}{-9} \times (-9) = (-5) \times (-9)\), simplifying might look like:
  • Cancelling out numbers, as we did with \(-9\) to leave \( n \) alone.
  • Calculating the result of any multiplication or division left in the equation. For example, computing \(-5 \times -9 = 45\).
  • Double-checking the simplified equation to ensure it is correct.
Simplification brings the equation to a point where the answer is clear, ensuring all steps maintain the balance needed to solve for the variable successfully.

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Most popular questions from this chapter

Solve each equation. $$4(x-7)=5(x+2)$$

Solve each equation. $$-2(x-4)-(3 x-2)=-2+(-6 x+2)$$

Solve each equation. (a) \(-2(x-1)=-2 x+2\) (b) \(3(x+4)=3 x-4\) (c) \(5(x-1)=-5 x-5\) (d) \(\frac{x-3}{3}+4=3\) (e) \(\frac{x+2}{3}+1=\frac{x-2}{3}\) (f) \(\frac{x-1}{5}-2=\frac{x-11}{5}\) (g) \(4(x-2)-2(x+3)=2(x+6)\) (h) \(5(x+3)-3(x-5)=2(x+15)\) (i) \(7(x-1)+4(x-2)=15(x-1)\)

Solve each equation. $$\frac{3}{4} x-\frac{2}{3}=\frac{5}{6}$$

Solve each equation. $$-2(x+6)+3(3 x-2)=-3(x-4)$$
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