Section 4.5 - Solving A Linear System Graphically

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• 1.

Determine the point of intersection of the lines y = 3x + 5  and  2x + y = 10

• A.

(3, 2)

• B.

(-5, 20)

• C.

(0, 5)

• D.

(1, 8)

D. (1, 8)
Explanation
To determine the point of intersection of the two lines, we can solve the system of equations formed by the two lines. By substituting the value of y from the first equation into the second equation, we get 2x + 3x + 5 = 10. Simplifying this equation gives us 5x + 5 = 10. Solving for x, we find that x = 1. Substituting this value of x back into the first equation, we can solve for y and find that y = 8. Therefore, the point of intersection is (1, 8).

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• 2.

The school store charges \$10 per shirt.  The store paid the supplier \$3 per shirt plus a one-tome set up fee of \$25.  Write a system of equations that represents this situation.

• A.

Y = 10x y = 3x + 25

• B.

Y = 10 y = 3x + 25

• C.

Y = 10x y = 3x

• D.

Y = 7x + 25 y = 5x

A. Y = 10x y = 3x + 25
Explanation
The equation y = 10x represents the price at which the school store sells each shirt, with y being the price and x being the number of shirts. The equation y = 3x + 25 represents the cost to the store to purchase each shirt from the supplier, with y being the cost and x being the number of shirts. By setting up these two equations, we can represent the situation where the store charges \$10 per shirt and pays \$3 per shirt plus a one-time set up fee of \$25 to the supplier.

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• 3.

Determine the point of intersection of the lines y = 4 and x = 3

• A.

Infinitely many

• B.

None

• C.

(3, 4)

• D.

(4, 3)

C. (3, 4)
Explanation
The given lines are y = 4 and x = 3. The line y = 4 is a horizontal line passing through the y-coordinate 4. The line x = 3 is a vertical line passing through the x-coordinate 3. These two lines intersect at the point (3, 4), as the x-coordinate of the intersection point is 3 and the y-coordinate is 4. Therefore, the correct answer is (3, 4).

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• 4.

What type of lines will have no points of intersection?

• A.

Parallel

• B.

Perpendicular

• C.

Intersecting

• D.

Coinciding

A. Parallel
Explanation
Parallel lines will have no points of intersection because they are always equidistant and never meet, no matter how far they are extended. This is a fundamental property of parallel lines in Euclidean geometry.

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• 5.

Consider the relation x + 2y = 6.  Which of the following relations will intersect this relation at (2, 2)?

• A.

6x + y = 2

• B.

2x + y = 4

• C.

X + y = 4

• D.

X + y = 2

C. X + y = 4
Explanation
The given relation is x + 2y = 6. To find the relation that intersects this at (2, 2), we substitute x = 2 and y = 2 into each of the answer choices.

For 6x + y = 2, substituting x = 2 and y = 2 gives 12 + 2 = 2 which is not true.

For 2x + y = 4, substituting x = 2 and y = 2 gives 4 + 2 = 4 which is not true.

For x + y = 4, substituting x = 2 and y = 2 gives 2 + 2 = 4 which is true.

For x + y = 2, substituting x = 2 and y = 2 gives 2 + 2 = 2 which is not true.

Therefore, the relation x + y = 4 is the correct answer as it intersects the given relation at (2, 2).

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• 6.

Find the point of intersection for the lines 3x - 2y = 5 and 6x - 4y = 6

• A.

None

• B.

Infinitely many

• C.

(0, 0)

• D.

(0, -4)

A. None
Explanation
The given lines 3x - 2y = 5 and 6x - 4y = 6 are parallel to each other, meaning they will never intersect. Therefore, the correct answer is "none".

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• 7.

A-1 plumbing charges \$75 plus \$50/h.  Plumbing Masters charges \$100 plus \$25/h.  At what point is the charge the same for the two companies?

• A.

2h

• B.

1h

• C.

1/2 h

• D.

3h

B. 1h
Explanation
The charge for A-1 plumbing is \$75 plus \$50 per hour, while Plumbing Masters charges \$100 plus \$25 per hour. To find the point at which the charges are the same for both companies, we need to set up an equation. Let x be the number of hours. The equation would be 75 + 50x = 100 + 25x. Simplifying this equation, we get 25x = 25, which means x = 1. Therefore, the charge is the same for both companies after 1 hour.

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• 8.

Cheap Cell Phones charges no flat fee and \$0.25 per minute.  Basic Cell Phones charges a flat fee of \$15 plus \$0.10 per minute.  What advice would you give someone who was deciding which company to choose?

• A.

Basic Cell Phones is always more expensive, so choose Cheap Cell Phones

• B.

Both plans cost the same, so wither choice is good

• C.

If the person is going to use less than 100 min, choose Basic Cell Phones. Otherwise choose Cheap Cell Phones

• D.

If the person is going to use less than 100 min, choose Cheap Cell Phones. Otherwise choose Basic Cell Phones

D. If the person is going to use less than 100 min, choose Cheap Cell Phones. Otherwise choose Basic Cell Phones
Explanation
If the person is going to use less than 100 minutes, it would be more cost-effective to choose Cheap Cell Phones since they do not have a flat fee and charge \$0.25 per minute. However, if the person is going to use more than 100 minutes, Basic Cell Phones would be a better option as they have a flat fee of \$15 and charge \$0.10 per minute.

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• 9.

Determine the point of intersection of the lines x + y = 8 and y = x + 4

• A.

None

• B.

Infinitely many

• C.

(2, 6)

• D.

(4, 4)

C. (2, 6)
Explanation
To determine the point of intersection of the lines x + y = 8 and y = x + 4, we can substitute the value of y from the second equation into the first equation. By doing so, we get x + (x + 4) = 8, which simplifies to 2x + 4 = 8. Solving for x, we find x = 2. Substituting this value back into the second equation, we get y = 2 + 4 = 6. Therefore, the point of intersection is (2, 6).

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• 10.

• A.

20 cans

• B.

10 cans

• C.

5 cans

• D.

1 can