How to find the solution to an equation

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SAT Math › How to find the solution to an equation

Questions 1 - 10

Pets Plus makes bird houses. Their monthly fixed expenses are $750. The cost for each bird house is $15. The bird houses sell for $40.

If Pets Plus sells 50 bird houses, what is the profit?

$\$500$

$\$250$

$\$750$

$\$300$

$\$625$

Explanation

Let $x$ = the number of birdhouses sold each month.

$Revenue=40x$

$Costs=15x+750$

$Profit = Revenue-Costs$

$=40x-15x-750$

$=25x-750$

Substituting in 50 for $x$ gives an answer of 500, so the profit on 50 birdhouses is $500.

If 5 + x is 5 more than 5,what is the value of 2_x_?

Explanation

5 more than 5 = 10

5 + x = 10

Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10

Two cars start 25 mile apart and drive away from each other in opposite directions at speeds of 50 and 70 miles per hour. In approximately how many minutes will they be 400 miles apart?

None of the other answers

187.5

200

3.33

3.125

Explanation

The cars have a distance from each other of 25 + 120t miles, where t is the number of hours, 25 is their initial distance and 120 is 50 + 70, or their combined speeds. Solve this equation for 400:

25 + 120t = 400; 120t = 375; t = 3.125

However, the question asked for minutes, so we must multiply this by 60:

3.125 * 60 = 187.5 minutes.

Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?

Explanation

In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.

Three consecutive even numbers add to 42. What is the middle number?

$14$

$10$

$12$

$16$

$18$

Explanation

Let $x$ = 1st even number, $x+2$ = 2nd even number, and $x+4$ = 3rd even number.

Then the equation to solve becomes $x+(x+2)+(x+4)=42$ .

$3x+6=42$

Thus $x=12,x+2=14,\ and\ x+4=16$ , so the middle number is 14.

Translate into an algebraic expression:

One third of the difference of a number and 3 equals five times the number.

$\frac{1}{3}\left ( x-3 \right )=5x$

$\frac{1}{3}x-3=5$

$\frac{1}{3}x-3=5x$

$\frac{1}{3}\left ( x+3 \right )= 5x$

$3\left ( x-3 \right )=5x$

Explanation

You have to form the difference first and must be enclosed in parentheses which gets multiplied by 1/3. The word "equals" means the "=" symbol

Which of the following equations has as its solution set $\left \{ -18.5, 10.5 \right \}$ ?

Explanation

The absolute value of a nonnegative number is the number itself; the absolute value of a negative number is its positive opposite.

By substitution, 10.5 can be seen to be a solution of each of the equations in the five choices. For example:

$|2 \cdot 10.5 + 6| = 27$

- a true statement.

That 10.5 is a solution of the other four equations can be proved similarly. Therefore, the question is essentially to choose the equation with as its other solution. Again, we can do this using substitution in each equations. We see that is correct as follows:

- a true statement.

Similar substitution in the other four statements shows that is not a solution of any of them; for example, in :

- a false statement.

The Widget Company makes widgets. The monthly fixed costs are $750. It costs $45 to make each widget. The widgets sells for $75 a piece.

The Widget Company wants to make a profit of $3,000. How many widgets must be sold?

Explanation

Profits = Revenues - Costs

Revenue:

Costs:

Profit:

So the equation to solve becomes

So a $3,000 profit occurs when they sell 125 widgets

Give the solution set.

$x = -16\frac{1}{2}$ or $x = 24\frac{1}{2}$

$x = 16\frac{1}{2}$ or $x = 24\frac{1}{2}$

$x = -16\frac{1}{2}$ or $x =- 24\frac{1}{2}$

$x = 16\frac{1}{2}$ or $x =- 24\frac{1}{2}$

The equation has no solution.

Explanation

Either or . Solve them separately:

$\frac{-4x}{-4}= \frac{-98}{-4}$

$x = 24\frac{1}{2}$

$\frac{-4x}{-4}= \frac{66}{-4}$

$x = -16\frac{1}{2}$

James is an artist and makes number of portraits per month. At this rate, how many months will it take him to paint portraits?

$\frac{t}{p}$

$\frac{100p}{t}$

$\frac{p}{t}$

Explanation

To figure out the total portraits done by James, you must multiply rate by the number of months . This equation would be . To solve for the number of months you must divide both sides by . This gives us $m=\frac{t}{p}$ .

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