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Chapter 3: Problem 83

Solve each equation. $$ 9-x=-4 $$

Short Answer

Expert verified
The solution to the equation is \( x = 13 \).

Step by step solution

01

Isolate the Variable Term

To isolate the variable term, add \(-9\) to each side of the equation to cancel out the \(9\) on the left side.\[ 9 - x - 9 = -4 - 9 \]
02

Simplify Both Sides

Simplify both sides of the equation:\[ -x = -13 \]
03

Solve for the Variable

To solve for \(x\), multiply both sides of the equation by \(-1\) to get rid of the negative sign in front of \(x\):\[ -x \times -1 = -13 \times -1 \]\[ x = 13 \]

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Key Concepts

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

Isolate the Variable
To solve an equation, one core concept is isolating the variable. This means getting the variable (like x) by itself on one side of the equation. In our example, we start with: $$ 9 - x = -4 $$ The first step is to move all terms that don't have the variable to the other side. Here, we can subtract 9 from both sides: \[ 9 - x - 9 = -4 - 9 \] This simplifies to: \[ -x = -13 \] Now, the variable term is isolated on one side of the equation.
Simplify Both Sides
Once the variable is isolated, we need to simplify both sides of the equation. Simplification involves performing any arithmetic operations to reduce each side to its simplest form. After isolating the variable term, our equation looks like this: \[ -x = -13 \] At this point, there are no further operations we can perform to simplify this further.
Multiplication Property of Equality
Another important concept is the multiplication property of equality. This property states that if you multiply both sides of an equation by the same number, the two sides remain equal. In our case, we have the equation: \[ -x = -13 \] To solve for x, we need to get rid of the negative sign in front of the variable. We do this by multiplying both sides of the equation by -1: \[ -x \times -1 = -13 \times -1 \] This simplifies to: \[ x = 13 \] By using the multiplication property of equality, we have successfully solved for the variable.

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Most popular questions from this chapter

Solve each problem. Suppose that it costs \(\$ 5000\) to start up a business selling snow cones. Furthermore, it costs \(\$ 0.50\) per cone in labor, ice, syrup, and overhead. Then the cost to make \(x\) snow cones is given by \(y\) dollars, where $$ y=0.50 x+5000 $$ Express each of the following as an ordered pair. (a) When 100 snow cones are made, the cost is \(\$ 5050\). (b) When the cost is \(\$ 6000\), the number of snow cones made is 2000 .

For each pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither. See Example \(6 .\) $$ \begin{aligned} &5 x-3 y=-2\\\ &3 x-5 y=-8 \end{aligned} $$

Graph each linear equation. \(x=-y+6\)

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 2 x=y-4 $$

Graph each linear equation. \(y=-1\)
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