# Algebra 1 : How to write expressions and equations

## Example Questions

### Example Question #31 : How To Write Expressions And Equations

Write an equation based on the following sentence:

Eight minus three times a number is 4 less than 6 times that number.

Explanation:

Let  represent the number.

"Eight minus three times a number" can be written as: .

"is" means equals or "".

"4 less than 6 times that number" can be understood to mean 6 times a number minus 4, or .

Putting these statements together gives:

### Example Question #32 : How To Write Expressions And Equations

Rewrite the expression in simplest terms.

Explanation:

When we combine like terms, we have to always first follow the order of operations (PEMDAS = parentheses, exponents, multiplication, division, addition, subtraction). So, we must first simplify what is inside the parentheses, then the exponents, the multiplication and division, then addition and subtraction. Don't forget to simplify like terms!

### Example Question #33 : How To Write Expressions And Equations

Rewrite the expression in simplest terms.

Explanation:

Here is the expression given: .

To simplify, follow the order of operations.

Distribute through the terms in the inner parentheses:

Now distribute  into the terms of the remaining parentheses. Remember that  multiplied by itself produces , but  multiplied by  produces :

Complete the multiplication to finish expanding:

### Example Question #34 : How To Write Expressions And Equations

Mike owns a bakery that specialized in artisan bread. When Mike doesn't sell any loaves, he loses $15 due to cost of materials, but each loaf of bread sold generates$3 profit.

Write out the equation for Mike's bakery profits in slope-intercept form, where:

and

Explanation:

The instructions tell us that the x-axis represents the number of loaves of bread sold and the y-axis represents net profit.

Based upon this information, the $15 loss in profit when 0 loaves of bread are sold is the y-intercept of the equation, beacuse it represents the value of f(x=0). So far, we know the y-intercept of our equation: y = _x - 15 The word problem mentions that each loaf of bread sold generates$3 income. This represents the rise in the y-value in respect to the increase in the x-value, meaning this value is the slope of our equation:

y = 3x - 15

### Example Question #35 : How To Write Expressions And Equations

There is a perfect linear correlation between the number of people (in thousands) that visit Golden Gate Park daily and the outside temperature in degrees Farenheit. Looking at the below diagram, create a linear equation that the Department of Parks and Recreation can use to estimate the number of park visitors, based upon the temperature.

Explanation:

The first step to finding the corresponding linear equation is to find the slope:

The next step is finding the y-intercept:

Using the formula for calculating the slope, we can isolate the unknown Y-value where X=0

Therefore, the equation is in slope-intercept form and comes out as:

### Example Question #36 : How To Write Expressions And Equations

To reduce pollution the San Francisco Bay Area, the state government offers a cash rebate to manufacturing plants who can reduce the number of metric tons of sulfur they emit annually. For every metric ton of sulfur that a factory does NOT emit, they will recieve a cash rebate of $100. However, if the factory emits MORE sulfur than they normally do, then they are fined$100 per each additional metric ton.

Provided with this information, create a slope-intercept equation, where  represents the DIFFERENCE in metric tons of pollution, and  represents the amount of the rebate/fine.

Explanation:

Since factories will not earn nor pay any money for emitting the same number of metric tons of sulfur as usual, that tells us that the Y-Intercept is 0. This is because when the value for Y=0, the value for X is also 0.

The next step is finding the slope of the equation.

The Y-Value represents the cash rebate/fine amount, while the X-Value represents the difference in number of metric tons of sulfur emitted.

The slope is the rise (Y-Value) divided by the run (X-Value):

Since we now have our Slope and our Y-Intercept, we can conclude that our equation is:

### Example Question #632 : Intermediate Single Variable Algebra

(9x2 – 1) / (3x – 1) =

3x

3x – 1

(3x – 1)2

3

3x + 1

3x + 1

Explanation:

It's much easier to use factoring and canceling than it is to use long division for this problem. 9x2 – 1 is a difference of squares. The difference of squares formula is a2 – b2 = (a + b)(a – b). So 9x2 – 1 = (3x + 1)(3x – 1). Putting the numerator and denominator together, (9x2 – 1) / (3x – 1) = (3x + 1)(3x – 1) / (3x – 1) = 3x + 1.

### Example Question #371 : Mathematical Relationships And Basic Graphs

Simplify the following equation.

Explanation:

We can simplify the natural log exponents by using the following rules for naturla log.

Using these rules, we can perform the following steps.

Knowing that the e cancels the exponential natural log, we can cancel the first e.

Distribute the square into the parentheses and calculate.

Remember that a negative exponent is equivalent to a quotient. Write it as a quotient and then you're finished.

### Example Question #5 : Imaginary Numbers

Identify the real part of

none of the above.

Explanation:

A complex number in its standard form is of the form: , where  stands for the real part and  stands for the imaginary part. The symbol  stands for .

The real part in this problem is 1.

### Example Question #37 : How To Write Expressions And Equations

Mr. Wiggins does not have enough books for all the students in his algebra class. He has 20 students, 10 textbooks, and 16 workbooks. He wants to divide his students into work groups according to the following rules:

- Every student must work in a group of 2 (a pair) or 3 (a trio).

- Every Pair or Trio must have at least 1 textbook and at least 2 workbooks.

How many pairs and trios should Mr. Wiggins divide his students into if he wants to have as many pairs as possible while following these rules?

pairs, trios

pairs, trios

pair, trios

pairs, trios

pairs, trios

pairs, trios

Explanation:

The conditions of Mr. Wiggins's problem can be expressed with an inequality and an equation to narrow down the number of pairs and trios that meets his conditions.

First, since each pair or trio must have at least workbooks, dividing the  total workbooks into sets of  means that there cannot be more than pairs and trios total. This can be expressed in the following inequality where  is the number of pairs and  is the number of trios.

Since each student must be in a pair or trio, we can set the number of total students () equal to the number of students in pairs () plus the number of students in trios ().

To satisfy this inequality and this equation while maximizing  (since Mr. Wiggins wants "as many pairs as possible"), start by substituting the maximum number of pairs () for , then work downward.

pairs would mean trios, according to the inequality . This does not satisfy the equation . Essentially, even though all of the workbooks are used, not all of the students have been accounted for. Continuing this pattern for , the sets , , , and  satisfy the inequality, but they do NOT satisfy the equation.

satisfies the inequality and the equation, so the answer is pairs and trios.