Algebra - How to write expressions and equations

Solve the following system of linear equations by the elimination method:

$\left ( 0, -1 \right )$

$\left ( 0, 1 \right )$

$\left ( 1, 0 \right )$

$\left ( 1, -1 \right )$

$\left ( -1, 0 \right )$

Explanation

We would like to eliminate .

Hence we multiply the first equation by which gives us the following equations:

Adding the above two equations eliminates the variable . We are left with:

and so

Replacing with in the original equation gives us

$x - 3\left ( -1 \right ) = 3$

solving for gives us

Hence the solution is $\left ( 0, -1 \right )$ .

Which of the following sentences translates to the algebraic equation ?

Eight multiplied by the difference of a number and three is equal to forty.

Three subtracted from the product of eight and a number is equal to forty.

The product of eight and a number subtracted from three is equal to forty.

The product of three and a number less eight is equal to forty.

Eight multiplied by the difference of three and a number is equal to forty.

Explanation

The expression on the left shows eight being multiplied by an expression in parentheses; that expression is the difference of an unknown number and three. The whole right expression is therefore worded as "Eight multiplied by the difference of a number and three"; the equality symbol and the forty round out the answer.

(9_x_2 – 1) / (3_x_ – 1) =

3_x_ + 1

3_x_ – 1

(3_x_ – 1)2

3_x_

Explanation

It's much easier to use factoring and canceling than it is to use long division for this problem. 9_x_2 – 1 is a difference of squares. The difference of squares formula is a_2 – b_2 = (a + b)(a – b). So 9_x_2 – 1 = (3_x + 1)(3_x – 1). Putting the numerator and denominator together, (9_x_2 – 1) / (3_x_ – 1) = (3_x_ + 1)(3_x_ – 1) / (3_x_ – 1) = 3_x_ + 1.

Rewrite the expression in simplest terms, where is the imaginary number $\sqrt-1$ .

$i* \sqrt{-12}$

$-2\sqrt3$

$2\sqrt3$

$-\sqrt{12}$

Explanation

Writing this expression in simplest terms can be achieved by first factoring the radical into its smallest factors.

$i*\sqrt{-12}=\sqrt{-1} * (\sqrt{-1}\sqrt{4}\sqrt{3})$

Multiplying the two $\sqrt{-1}$ together results in . Multiplying this by $\sqrt4\sqrt3$ (which is simplified to $2\sqrt3$ ) results in the answer $-2\sqrt3$ .

$\sqrt{-1}*(\sqrt{-1}\sqrt{4}\sqrt{3})=(\sqrt{-1})^2*2\sqrt{3}=-2\sqrt{3}$

Translate the following sentence into an algebraic expression:

"Nine plus the quotient of an unknown number and six"

$9+\frac{x}{6}$

$9-\frac{x}{6}$

$9+\frac{6}{x}$

$9-\frac{6}{x}$

None of the above

Explanation

In order to translate the above sentence into an algebraic expression, let's consider each part of the sentence:

"Nine plus the quotient of an unknown number and six"

"Nine plus..." means that we're going to take the sum of and everything that follows in the sentence.

"The quotient of an unknown number and six" means the value we get when we divide an unknown number by , or $\frac{x}{6}$ .

Putting all of the above together, we get $9+\frac{x}{6}$ .

Translate the following sentence into an algebraic expression:

"The square root of the sum of 5 and twice an unknown number"

$\sqrt{5+2x}$

$\sqrt{5-2x}$

$\frac{1}{\sqrt{5+2x}}$

$-\frac{1}{\sqrt{5+2x}}$

Explanation

In order to translate the above sentence into an algebraic expression, let's consider each part of the sentence:

"The square root of the sum of 5 and twice an unknown number"

"The square root of..." means that we are going to take the square root, $\sqrt{}$ , of the the expression that follows.

"The sum of ..." means that we're going to add one or more numbers to .

"Twice an unknown number" means that we're going to multiply times an unknown number that, in this instance, we'll call . We can call this product .

Putting all of the above together, we get $\sqrt{5+2x}$ .

Translate the following sentence into an algebraic expression:

"The difference between 7 and the quotient of an unknown number and 3"

$7-\frac{x}{3}$

$\frac{x}{3}-7$

$\frac{7-x}{3}$

$\frac{x-7}{3}$

$\frac{7}{3}x$

Explanation

In order to translate the above sentence into an algebraic expression, let's consider each part of the sentence:

"The difference between 7 and the quotient of an unknown number and 3"

"The difference between 7 and..." means that we are subtracting the remainder of the expression from 7, or ...

"The quotient of an unknown number and 3" means that we are dividing an unknown number - let's call it - by , or $\frac{x}{3}$ .

Putting all of the above together, we get $7-\frac{x}{3}$ .

Express seventeen times .

$\frac{17}{w}$

$w^{17}$

Explanation

When we see times, we are dealing with multiplication. It doesn't matter which comes first since multiplication is commutative. Our expression is .

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.

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

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

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:

$\fn_cm Simplify ; x^{3} + 3x^{2} - 6x + 10 -(x^{2} +x + 13)$

$\fn_cm -3-7x+2x^2+x^3$

Explanation

$Simplify ; x^{3} + 3x^{2} - 6x + 10 -(x^{2} +x + 13)$
$\fn_cm Distribute ; -1 ; over ; x^2+x+13.$ $-(x^2+x+13) \rightarrow ; -x^2-x-13$ $\fn_cm \fn_cm -(x^2+x+13) \rightarrow ; =-x^2-x-13$

$\fn_cm \rightarrow x^3+(3 x^2-x^2)+(-6 x-x)+(10-13)$ $\fn_cm \fn_cm Combine ; like ; terms ; 3 x^2-x^2\rightarrow2x^2$