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Algebra 1 : Linear Equations

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Example Questions

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.

Possible Answers:

$y=\frac{-1}{10}x-1$

$y=\frac{1}{10}x+1$

$y=\frac{1}{10}x-1$

$y=\frac{-1}{10}x+1$

$y=\frac{1}{10}x+6$

Correct answer:

$y=\frac{1}{10}x+1$

Explanation:

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

$Slope=\frac{Rise}{Run}=\frac{8.5-8}{75-70}=\frac{0.5}{5}=\frac{1}{10}$

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

Y=mX+b

$Slope=\frac{Rise}{Run}=\frac{Y_{n}-Y_{m}}{X_{n}-X_{m}}=\frac{8.5-Y_{0}}{75-0}=\frac{1}{10}$

$\frac{1}{10}=\frac{8.5-Y_{0}}{75}$

$\frac{7.5}{75} = \frac{1}{10}$

$\frac{8.5-Y_{0}}{75}=\frac{7.5}{75}$

$Y_{0}=1$

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

$Y=\frac{1}{10}X+1$

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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.

Possible Answers:

y=-100x

$y=\frac{1}{100}x$

y=100x

$y=-\frac{-1}{100}x$

Correct answer:

y=-100x

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):

$\frac{Rise}{Run}=\frac{100 Dollar Rebate}{1LessTonOfSulfur}=\frac{100}{-1}=-100$

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

Y=-100X+0

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Example Question #951 : Algebra 1

(9x² – 1) / (3x – 1) =

Possible Answers:

3x – 1

3x + 1

(3x – 1)²

Correct answer:

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.

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Example Question #125 : Logarithms

Simplify the following equation.

$(e^{(\ln2+\ln93-\ln6)}e^{-1})^2$

Possible Answers:

31e

$\frac{31}{e^{-2}}$

961e^2

$\frac{961}{e^2}$

$\frac{31}{e}$

Correct answer:

$\frac{961}{e^2}$

Explanation:

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

$\ln A+\ln B=\ln AB$

$\ln A - \ln B=\ln\frac{A}{B}$

Using these rules, we can perform the following steps.

$(e^{(\ln2+\ln93-\ln6)}e^{-1})^2$

$(e^{(\ln(2(93))-\ln6)}e^{-1})^2$

$(e^{(\ln(\frac{2(93)}{6})}e^{-1})^2$

$(e^{\ln 31}e^{-1})^2$

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

$(31e^{-1})^2$

Distribute the square into the parentheses and calculate.

$(31)^2(e^{-1})^2=961e^{-2}$

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

$961e^{-2}=\frac{961}{e^2}$

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Example Question #2 : Imaginary Numbers

Identify the real part of 1-3i

Possible Answers:

none of the above.

Correct answer:

Explanation:

A complex number in its standard form is of the form: a+bi , where stands for the real part and stands for the imaginary part. The symbol stands for $\sqrt{-1}$ .

The real part in this problem is 1.

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Example Question #951 : Linear 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?

Possible Answers:

pair, trios

pairs, trios

Correct answer:

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.

$p + t \leq 8$

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 ().

2p + 3t = 20

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 $p + t \leq 8$ . This does not satisfy the equation 2p + 3t = 20 . Essentially, even though all of the workbooks are used, not all of the students have been accounted for. Continuing this pattern for (p,t) , the sets (8,0) , (7,1) , (6,2) , and (5,3) satisfy the inequality, but they do NOT satisfy the equation.

(4,4) satisfies the inequality and the equation, so the answer is pairs and trios.

$4+4\leq 8\rightarrow true$

$2(4)+3(4)=20\rightarrow true$

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Example Question #952 : Linear Equations

What is the mathematical representation of the square root of the quantity of two x minus three?

Possible Answers:

$2x-\sqrt{3}$

$\sqrt{2}x-3$

$\sqrt{2x-3}$

$2\sqrt{x-3}$

$\sqrt{2}x-3$

Correct answer:

$\sqrt{2x-3}$

Explanation:

When representing mathematical expressions with numbers and symbols, the phrasing is key.

"the quantity of" means to group the next description.

The quantity of two x minus three : 2x-3

"Square root of" means to draw the square root symbol around the next description or to give the description the exponent of $\frac{1}{2}$ .

$\sqrt{2x-3}=(2x-3)^{1/2}$

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Example Question #953 : Linear Equations

Simplify the following expression.

$\frac{2x^{4}y^{-3}z^{-2}\cdot 5x^{-3}y^{6}z\cdot x^{5}z^{3}}{8x^{-2}y^{4}z\cdot y^{3}z^{2}}$

Possible Answers:

$\frac{5}{x^{120}y^{-\frac{3}{2}}z^{12}}$

$\frac{5x^{4}}{4y^{4}z}$

$\frac{5x^{8}}{4y^{4}z}{}$

$\frac{5x^{30}y^{10}}{4z^{12}}$

$\frac{5(xyz)^{3}}4{}$

Correct answer:

$\frac{5x^{8}}{4y^{4}z}{}$

Explanation:

To simplify, cancel like terms and apply exponent rules.

Remember when like bases are being multiplied together you add their exponents. Likewise, if you are dividing like bases you subtract the exponent that is in the denominator from the exponent that is in the numerator.

$\frac{2x^{4}y^{-3}z^{-2}\cdot 5x^{-3}y^{6}z\cdot x^{5}z^{3}}{8x^{-2}y^{4}z\cdot y^{3}z^{2}}$

$\\=\frac{10x^{6}y^{3}z^{2}}{8x^{-2}y^{7}z^{3}}\\ \\=\frac{5x^{8}}{4y^{4}z}$

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Example Question #954 : Linear Equations

Express three more than .

Possible Answers:

x-3

$\frac{3}{x}$

$\frac{x}{3}$

x+3

Correct answer:

x+3

Explanation:

Take every word and translate into math.

more than means that you need to add to something.

That something is so just combine them to have an expression of x+3 .

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Example Question #952 : Algebra 1

Express product of five and .

Possible Answers:

$\frac{y}{5}$

5+y

5-y

y-5

Correct answer:

Explanation:

Take every word and translate into math.

Product is multiplication.

So we have as the answer.

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Algebra 1 : Linear Equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #35 : How To Write Expressions And Equations

Example Question #36 : How To Write Expressions And Equations

Example Question #951 : Algebra 1

Example Question #125 : Logarithms

Example Question #2 : Imaginary Numbers

Example Question #951 : Linear Equations

Example Question #952 : Linear Equations

Example Question #953 : Linear Equations

Example Question #954 : Linear Equations

Example Question #952 : Algebra 1

All Algebra 1 Resources

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