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

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Example Questions

Example Question #21 : How To Find The Solution For A System Of Equations

The sum of the digits in a positive three-digit whole number is equal to 13. When the digits are reversed, the number's value is decreased by 594. If the first digit must be a prime number, then what is the sum of the squares of the three digits?

Possible Answers:

Correct answer:

Explanation:

Since the number has three digits, there must be a digit in the hundreds place, a digit in the tens place, and a digit in the ones place. Let's let h represent the digit in the hundreds place, t represent the digit in the tens place, and let u represent the digit in the ones place. Thus, we can represent our mystery number as follows:

mystery number = 100h + 10t + u

First, we are told that the sum of the digits is 13. Thus, we can write the following equation.

h + t + u = 13

Next, we are told that reversing the digits decreases the value of the original number by 594. We already established that the original number can be represented as 100h + 10t + u. We can then represent the reversed number as 100u + 10t + h. We can now write the following equation:

original number = 594 + reversed number

100h + 10t + u = 594 + 100u + 10t + h

Let's get the h, t, and u terms on the same side. Subtract h from both sides.

99h + 10t + u = 594 + 100u + 10t

Subtract 10t from both sides.

99h + u = 594 + 100u

Subtract 100u from both sides.

99h – 99u = 594

Since 99 and 594 are divisible by 9, we will divide both sides by 9.

11h – 11u = 66

Divide both sides by 11.

h – u = 6

We are told that the first digit, represented by h, must be prime. Let's solve for h in terms of u by adding u to both sides.

h = 6 + u

Since h is a digit, it can't be larger than 9. (The possible values for a digit are 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.) The only prime numbers less than 9 are 2, 3, 5, and 7. However, if h were 2, 3, or 5, then u would be negative, and we can't have a negative digit. In short, the only value that h can be is 7.

7 = 6 + u

Since h is 7, u must be 1.

If h = 7 and u = 1, we will solve for t by using the first equation h + t + u = 13.

7 + t + 1 = 13

Subtract 8 from both sides.

t = 5

Since h = 7, t = 5, and u = 1, the number must be 751. We are told to find the squares of the digits.

sum of squares of digits = 7² + 5² + 1² = 49 + 25 + 1 = 75.

The answer is 75.

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

Given the following system of inequalities, find the answer below that does NOT satisfy the system.

$\dpi{80} x-y> 1$

$\dpi{80} 2x-3y< 6$

Possible Answers:

$\dpi{80} (0,-1)$

$\dpi{80} (4,1)$

$\dpi{80} (1,-1)$

$\dpi{80} (3,1)$

$\dpi{80} (2,0)$

Correct answer:

$\dpi{80} (0,-1)$

Explanation:

To be a solution to the system, the point must satisfy BOTH inequalities. All five points satisfy the inequality $\dpi{80} 2x-3y< 6$ . On the other hand, $\dpi{80} \left (0, -1)$ does not satisfy $\dpi{80} x-y> 1$ .

$\dpi{80} \left ( 0) - (-1)> 1?$

$\dpi{80} 0+1 > 1?$

$\dpi{80} 1> 1? (No)$

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Example Question #21 : Systems Of Equations

20 fifth graders attended a birthday party. Each child got to choose between playing minature golf, which cost $4, or bowling, which cost $6. The total cost of the party was $96. How many children playing minature golf?

Possible Answers:

Correct answer:

Explanation:

Let x equal the number of children that played minature golf and let y equal the number of children who bowled.

Based on that, we can conclude that:

$x + y =20$

and

$4x + 6y =96$

Let's solve for x in the first equation by subtracting y from each side. $20-y=x$

Now, let's substitute that expression for x in the second equation:

$4(20-y) + 6y = 96$

Now, distribute the 4:

$80-4y + 6y =96$

Simplify:

$80+2y=96$

Subtract 80 from both sides:

$2y=16$

Divide both sides by 2:

$y=8$

Therefore, 8 children bowled. By substituting y into the first equation, we know that $x+8=20$

Solve for x by subtracting 8 from both sides: $x=12$

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Example Question #21 : How To Find The Solution For A System Of Equations

At what point will the lines $\dpi{100} \small 4x+2=y$ and $\dpi{100} \small 3x+3=y$ intersect?

Possible Answers:

(1, –6)

(6, –1)

(6, 1)

(1, 6)

(–1, 6)

Correct answer:

(1, 6)

Explanation:

In order to find this point, we must find the solution to the system of equations. we will use substitution, setting the two expressions for y equal to one another.

$\dpi{100} \small 4x+2=3x+3 \rightarrow x=1$

Then we plug this value back into either expression for y, giving us $\dpi{100} \small y=4\times 1+2=6$

So the point is (1, 6).

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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 $\frac{2}{17}$ .

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Example Question #22 : Systems Of Equations

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

Possible Answers:

(5,22)

(1,7)

(-4,13)

(22,5)

(11,27)

Correct answer:

(11,27)

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.

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Example Question #26 : How To Find The Solution For A System Of Equations

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:

Correct answer:

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.

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

$\small 4x+36-2x=48$

$\small 2x+36=48$

$\small 2x=12$

$\small x=6$

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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:

Correct answer:

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.

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Example Question #24 : 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.

What is the monthly break-even point?

Possible Answers:

Correct answer:

Explanation:

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

Costs: C(b)=600+35b

Revenue: R(b)=75b

So Costs = Revenue or 600+35b=75b

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

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Example Question #22 : 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.

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

Possible Answers:

Correct answer:

Explanation:

Profits = Revenue - Costs

Revenue: R(b)=75b

Costs: C(b)=600+35b

P(b)=R(b)-C(b)=75b-(600+35b)=40b-600

Solving:

200=40b-600

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

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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 #21 : How To Find The Solution For A System Of Equations

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

Example Question #21 : Systems Of Equations

Example Question #21 : How To Find The Solution For A System Of Equations

Example Question #25 : Systems Of Equations

Example Question #22 : Systems Of Equations

Example Question #26 : How To Find The Solution For A System Of Equations

Example Question #241 : Equations / Inequalities

Example Question #24 : Systems Of Equations

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

All SAT Math Resources

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