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SAT Math : How to find the solution for a system of equations

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Example Question #31 : Systems Of Equations

Solve the following system of equations:

2x + 3y = 19

3x + 2y = 16

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 $y = \frac{19}{3} - \frac{2}{3}x$ and thus $3x + 2(\frac{19 }{3}-\frac{2}{3}x) = 16$ and solving for x = 2 and then y = 5 .

So the sum of and is 7.

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Example Question #32 : Systems Of Equations

$4x+3z+8y = 26 \mbox{ and } x + 2y = 2$

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 (2x+y) :

4x+3z+8y = 4x + 8y + 3z = 4(x+2y) + 3z = 26

Substitution gives:

4(2)+3z =26

Subtract 8 from both sides:

3z = 18

Divide by 3:

z=6

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Example Question #33 : Systems Of Equations

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, :

120 =6r+e

Apply similar logic to Tom:

95=4r+e

Subtracting the second equation from the first equation results in:

25 = 2r

Divide both sides by 2:

r = 12.5

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:

95 = 4(12.5) +e

95 = 50 +e

Subtract 50 from both sides:

e = 45

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Example Question #2021 : Sat Mathematics

Solve the system of equations.

4x+6y=2

y=2x

Possible Answers:

$(\frac{2}{9}, \frac{1}{9})$

$(4,\frac{1}{4})$

$(\frac{1}{8}, \frac{1}{4})$

$(\frac{1}{8}, 4)$

Correct answer:

$(\frac{1}{8}, \frac{1}{4})$

Explanation:

4x+6y=2

y=2x

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.

4x+6y=2

4x+6(2x)=2

Now we can solve for .

4x+12x=2

16x=2

$x=\frac{2}{16}=\frac{1}{8}$

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

$y=2x\ \text{and}\ x=\frac{1}{8}$

$y=2(\frac{1}{8})$

$y=\frac{2}{8}=\frac{1}{4}$

The final answer will be the ordered pair $(\frac{1}{8}, \frac{1}{4})$ .

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Example Question #35 : Systems Of Equations

Solve for .

3y+6x=24

2y+x=16

Possible Answers:

4.8

Correct answer:

Explanation:

3y+6x=24

2y+x=16

Solve the system of equations using substitution.

First, isolate one of the variables. Since we are solving for , we are going to isolate in the second equation.

2y+x=16

x=-2y+16

Replace with -2y+16 in our first equation.

3y+6x=24

3y+6(-2y+16)=24

Now we can solve to isolate .

3y+(-12y+96)=24

-9y+96=24

-9y=-72

y=8

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Example Question #251 : Algebra

A train leaves the station going 60 miles per hour. Twenty minutes later another train leaves going 100 miles per hour. How much time it take from the time the second train leaves the station until it catches up with the first train?

Possible Answers:

30 minutes

42 minutes

28 minutes

10 minutes

20 minutes

Correct answer:

30 minutes

Explanation:

After 20 minutes the first train would have traveled 20 miles. Let x be the amount of time elapsed. When 20 + 60x = 100x you will have the time in hours. 20 = 40x, x = 0.5 hrs. 0.5 hrs = 30 minutes.

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Example Question #37 : Systems Of Equations

In the following system of equations, what is the value of m – n?

$-\frac{1}{2}m+n=3$

4m+2n=16

Possible Answers:

–2

Correct answer:

–2

Explanation:

Solve by method of elimination. Multiply the first equation by 8 to eliminate the variable, m. Our first equation will then become $\dpi{100} \small -4m+8n=24$ .

By adding this new equation

$\dpi{100} \small -4m+8n=24$

with our second equation

$\dpi{100} \small 4m+2n=16$

We will see that our m cancel out. We can now solve for n.

$\dpi{100} \small 10n=40$

$\dpi{100} \small n=4$

Now we have to plug in this value of n into any of our equations to find the value of m.

Let's use the second equation.

$\dpi{100} \small 4m+2\left (4 \right )=16$

$\dpi{100} \small 4m+8=16$

$\dpi{100} \small 4m=8$

$\dpi{100} \small m=2$

$\dpi{100} \small m-n=-2$

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Example Question #38 : Systems Of Equations

Solve for and .

$\dpi{100} \small 10x - y = 31$

$\dpi{100} \small y-x=5$

Possible Answers:

$\dpi{100} \small x=4, y=9$

$\dpi{100} \small x=2, y=7$

$\dpi{100} \small x=9, y=5$

$\dpi{100} \small x=7, y=2$

$\dpi{100} \small x=5, y=9$

Correct answer:

$\dpi{100} \small x=4, y=9$

Explanation:

Substitution needs to be used in order to solve this system of equations. From the second equation we know that $\dpi{100} \small y=5+x$ ,

Substitute that into the first equation and solve.

You get $\dpi{100} \small 10x - (5+x)=31$

$\dpi{100} \small 10x-5-x =31$

$\dpi{100} \small 9x =36$

$\dpi{100} \small x=4$

From there solve for y using the second equation.

$\dpi{100} \small y-x=5$

$\dpi{100} \small y-4=5$

$\dpi{100} \small y=9$

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Example Question #39 : Systems Of Equations

Solve for and .

4x+8y=12

5x+2y=15

Possible Answers:

$x=-3\ y=0$

None of these

$x=0\ y=-3$

$x=3\ y=0$

$x=0\ y=3$

Correct answer:

$x=3\ y=0$

Explanation:

4x+8y=12

5x+2y=15

Multiply the bottom equation by

Add the two equations together. This will allow you to cancel out from an equation. From here, you can proceed to solve for .

5x+2y=15

-20x-8y=-60

-16x=-48

x=3

From here, plug in the value into the equation of your choice to solve for .

4(3)+8y=12

y=0

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Example Question #40 : Systems Of Equations

If 2x + 3y = 10 and -x + 5y = 8 , then find the value of .

Possible Answers:

6.5

Correct answer:

Explanation:

We are essentially presented with a system of equations. To solve for y, we will need to solve the system. The easiest way to solve this particular system is by adding the equations together.

First, multiply the second equation by 2.

2(-x + 5y = 8)

-2x+10y=16

Adding the two equations together will allow you to cancel the x values and solve for y.

2x + 3y = 10

$+\ -2x+10y=16$

0x+13y=26

13y=26

y=2

If y equals 2, then 4y will be equal to 8.

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SAT Math : How to find the solution for a system of equations

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #31 : Systems Of Equations

Example Question #32 : Systems Of Equations

Example Question #33 : Systems Of Equations

Example Question #2021 : Sat Mathematics

Example Question #35 : Systems Of Equations

Example Question #251 : Algebra

Example Question #37 : Systems Of Equations

Example Question #38 : Systems Of Equations

Example Question #39 : Systems Of Equations

Example Question #40 : Systems Of Equations

All SAT Math Resources

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