SAT Math : Equations / Inequalities

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #25 : Systems Of Equations

Let \dpi{100} A be the point of intersection between the lines given by the equations \dpi{100} 2x-5y=4 and \dpi{100} 4x-7y=9.  What is the slope of the line that passes through \dpi{100} A and the origin?

Possible Answers:

-\frac{17}{6}

\frac{17}{6}

\frac{17}{2}

-\frac{17}{2}

\frac{2}{17}

Correct answer:

\frac{2}{17}

Explanation:

In order to solve this system of equations, it will probably be best to use the elimination method, in which we add or subtract two equations. In general, if \dpi{100} a=b, and \dpi{100} c=d, then \dpi{100} a+c=b+d. In other words, we can combine equations.

We need to solve the following equations:

2x-5y=4

4x-7y=9

We want to look for a way to combine these two equations that will eliminate at least one variable. If, for example, we multiplied the first equation by \dpi{100} -2 and then added this to the second equation, we could get rid of the \dpi{100} x terms. Let's multiply the first equation by \dpi{100} -2. Remember, we must multiply both sides.

-2(2x-5y)=(-2)(4)

Distribute the \dpi{100} -2.

-4x+10y=-8

We can now add -4x+10y=-8 to the equation 4x-7y=9. The sum of the left sides of each equation will equal the sum of the right sides.

(-4x+10y)+(4x-7y)=-8+9

We can simplify both sides. Notice that the \dpi{100} x terms on the left disappear, leaving only \dpi{100} y terms.

3y = 1

Divide both sides by 3.

y=\frac{1}{3}.

Now that we have \dpi{100} y, we can go back to either of our original two equations, substitute the value of \dpi{100} y we obtained, and solve for \dpi{100} x. Let's use the equation 2x-5y=4.

Substitute \dpi{100} \frac{1}{3} for \dpi{100} y.

2x-5(\frac{1}{3})=4

2x-\frac{5}{3}=4

Add \dpi{100} \frac{5}{3} to both sides. We will need to rewrite 4 with a denominator of 3.

2x = 4 + \frac{5}{3}=\frac{12}{3}+\frac{5}{3}=\frac{17}{3}

Divide by 2 on both sides. When dividing a fraction, we simply multiply by its reciprocal.

x = \frac{17}{3}\div 2=\frac{17}{3}\cdot \frac{1}{2}=\frac{17}{6}

Point \dpi{100} A, which is the intersection of the lines, must have the coordinates \left ( \frac{17}{6},\frac{1}{3} \right ).

The question ultimately asks for the slope of the line passing through \dpi{100} A and the origin. We can use the equation of the slope between two points with coordinates (x_{1},y_{1}) and (x_{2},y_{2}).

slope = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{\frac{1}{3}-0}{\frac{17}{6}-0}

= \frac{\frac{1}{3}}{\frac{17}{6}}

=\frac{1}{3}\div \frac{17}{6}=\frac{1}{3}\cdot \frac{6}{17}=\frac{2}{17}

The answer is \frac{2}{17}.

Example Question #22 : How To Find The Solution For A System Of Equations

Where do the lines y=3x-6 and y=2x+5 intersect?

Possible Answers:

Correct answer:

Explanation:

Finding the intersection of two lines is the same as solving simultaneous equations.  Here, both equations are already solved for , so the right sides of both equations are set equal to each other.  Solving for  results in a value of 11.  Setting  equal to 11 in either of the original equations gives a  value of 27.

Example Question #242 : Algebra

At a carnival, hamburgers cost $4 while hot dogs cost $2. Becky has to buy 18 items for her students. She spent a total of $48. How many hamburgers did she buy?

Possible Answers:

6

12

10

8

Correct answer:

6

Explanation:

Let x equal the number of hamburgers and y equal the number of hotdogs.

\small x+y=18

\small 4x+2y=48

Solve for y in the first equation.

\small y=18-x

Substitute that solution into the second equation.

\small 4x+2(18-x)=48

\small 4x+36-2x=48

\small 2x+36=48

\small 2x=12

\small x=6

Example Question #241 : Equations / Inequalities

This year, Amy is 3 times older than Jackie. In 6 years, Amy will be 2 times older than Jackie. How old is Jackie today?

Possible Answers:

6

12

18

10

Correct answer:

6

Explanation:

Let Amy's current age be equal to A and Jackie's age be equal to J.

Therefore, \small A=3J and \small (A+6)=2(J+6)

Substitute the first equation into the second.

\small (3J+6)=(2J+12)

\small J+6=12

\small J=6

Therefore, today Jackie is 6 years old.

Example Question #24 : How To Find The Solution For A System Of Equations

Bobbie's Boots makes winter boots.  Their monthly fixed expenses are $600.  The cost for making a pair of boots is $35.  The boots sell for $75 a pair.

What is the monthly break-even point?

Possible Answers:

Correct answer:

Explanation:

The break-even point is where the costs equal the revenue.

Costs: 

Revenue: 

So Costs = Revenue or

Solving for  shows the break-even point is achieved when 15 pairs of boots are sold in a month.

Example Question #21 : Systems Of Equations

Bobbie's Boots makes winter boots.  Their monthly fixed expenses are $600.  The cost for making a pair of boots is $35.  The boots sell for $75 a pair.

To make a profit of $200, how many pairs of boots must be sold?

Possible Answers:

Correct answer:

Explanation:

Profits = Revenue - Costs

Revenue: 

Costs: 

So

Solving:

 

for means that 20 pairs of boots must be sold to make $200 profit.

Example Question #2021 : Sat Mathematics

Solve the following system of equations:

What is the sum of  and ?

Possible Answers:

Correct answer:

Explanation:

This problem can be solved by using substitution.  Write the first equation  in terms of  and substitute it into the second equation.

So  and thus  and solving for  and then .

So the sum of  and  is 7.

Example Question #32 : Systems Of Equations

Find .

Possible Answers:

Correct answer:

Explanation:

We have 3 unkown variables and only 2 equations. Instead of trying to solve for   or , notice that we can substitute 2 in for the entire expression :

Substitution gives:

Subtract 8 from both sides: 

Divide by 3:

Example Question #2022 : Sat Mathematics

An amusement park charges both an entrance fee, and a fee for every ride. This fee is the same for all rides. Lisa went on 6 rides and paid 120 dollars. Tom went on only 4 rides and paid 95 dollars. What was the entrance fee?

Possible Answers:

Correct answer:

Explanation:

We need 2 equations, because we have 2 unkown variables. Let  = the entrance fee, and  = the fee per ride. One ride costs  dollars. We know that Lisa spent 120 dollars in total. Since Lisa went on 6 rides, she spent  dollars on rides. Her only other expense was the entrance fee,

Apply similar logic to Tom:

Subtracting the second equation from the first equation results in:

Divide both sides by 2:

So every ride costs 12.5 dollars. Plugging 12.5 back into one of the original equations allows us to solve for the entrance fee:

Subtract 50 from both sides:

Example Question #3 : New Sat Math No Calculator

Solve the system of equations.

Possible Answers:

Correct answer:

Explanation:

For this system, it will be easiest to solve by substitution. The  variable is already isolated in the second equation. We can replace  in the first equation with , since these two values are equal.

Now we can solve for .

Now that we know the value of , we can solve for  by using our original second equation.

The final answer will be the ordered pair .

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