Introduction to Systems of Equations

Show your work for each problem.

1.
What is a system of equations?
1. a)A graph with multiple lines
2. b)Two or more equations with the same variables
3. c)One equation with one variable
4. d)An expression with multiple terms
Answer: Two or more equations with the same variables
A system of equations is a set of two or more equations that share the same variables and must be solved together.
2.
What does the solution to a system of equations represent graphically?
1. a)The intersection point of the lines
2. b)The y-intercept of both lines
3. c)The slope of the lines
4. d)The distance between the lines
Answer: The intersection point of the lines
The solution to a system of two linear equations is the point where both lines intersect. This point satisfies both equations simultaneously.
3.
Is a solution to the system: and ?
1. a)Yes, it satisfies both equations
2. b)No, it satisfies neither equation
3. c)No, it only satisfies the second equation
4. d)No, it only satisfies the first equation
Answer: Yes, it satisfies both equations
Checking: (true) and (true). Since both equations are satisfied, is the solution.
4.
If and , what is the value of ?
Answer: 7
Adding the equations: , so , giving .
5.
If and , what is the value of ?
Answer: 3
From the previous problem, . Substituting into : , so .
6.
What happens when two lines are parallel?
1. a)The system has no solution
2. b)The system has infinitely many solutions
3. c)The system has exactly one solution
4. d)The system cannot be determined
Answer: The system has no solution
Parallel lines have the same slope but different y-intercepts. Since they never intersect, there is no point that satisfies both equations, so the system has no solution.
7.
Verify if is a solution to: and
Answer: Yes
1. Substitute and into the first equation . What do you get? 10
2. Is the first equation satisfied? (yes/no) yes
3. Substitute and into the second equation . What do you get? -1
4. Is the second equation satisfied? (yes/no) yes
8.
The lines and intersect at a point. What is the -coordinate of that point?
Answer: 2
Setting equal: . Adding to both sides: . Subtracting 1: . Dividing: .
9.
The lines and intersect at a point. What is the -coordinate of that point?
Answer: 5
From the previous problem, . Substituting into : .
10.
Find the solution to: and
Answer: (3, 5)
1. Add both equations together. What does equal? 3x
2. What does equal? 9
3. So . What is ? 3
4. Substitute into . What is ? 5
11.
At a store, 3 pencils and 2 erasers cost 8 euros. 1 pencil and 4 erasers cost 6 euros. Find the price of one pencil.
Answer: 2
1. Let = price of pencil, = price of eraser. Write the first equation. 3p + 2e = 8
2. Write the second equation. p + 4e = 6
3. From equation 2, express in terms of . p = 6 - 4e
4. Substitute into equation 1 and solve for . What is ? 1
5. Now find using . What is ? 2
12.
Which system has infinitely many solutions?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
The second equation simplifies to , which is identical to the first equation. Since both equations represent the same line, every point on that line is a solution.
13.
The sum of two numbers is 15 and their difference is 3. What is the larger number?
Answer: 9
Let = larger number, = smaller number. System: and . Adding: , so .
14.
A parking lot has cars and motorcycles. There are 50 vehicles in total and 160 wheels. How many cars are there?
Answer: 30
1. Let = cars, = motorcycles. Write the equation for total vehicles. c + m = 50
2. Cars have 4 wheels, motorcycles have 2. Write the wheel equation. 4c + 2m = 160
3. From equation 1, express in terms of . m = 50 - c
4. Substitute into the wheel equation and simplify. 4c + 2(50 - c) = 160
5. Solve for . How many cars are there? 30
15.
Find where the lines and intersect.
Answer: (2, 3)
1. From the second equation , express in terms of . x = y - 1
2. Substitute into . What equation do you get? 3(y - 1) + 2y = 12
3. Simplify and solve for . What is ? 3
4. Now find using . What is ? 2
16.
Movie tickets cost different amounts for adults and children. 2 adult tickets and 3 child tickets cost 36 euros. 3 adult tickets and 2 child tickets cost 39 euros. What is the price of one adult ticket in euros?
Answer: 9
System: and . Multiply first equation by 2: . Multiply second by 3: . Subtract: , so euros.