2024 MTAP SATURDAY PROGRAM IN MATHEMATICS GRADE 8 SESSION 3

SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES

A. Tell whether the given ordered pair is a solution to each system of linear equations.

1
𝑦=2 𝑦 =𝑥−1 𝑦 = 2𝑥 + 3 5𝑥 − 4𝑦 − 12 = 0
1. { 2. { 3. { 4. {
𝑥+𝑦 =1 𝑦 = 2𝑥 − 3 𝑥 + 3𝑦 = −1 2𝑥 + 2𝑦 = −3
1
(−1,2) (2,1) (−4,1) (2, − 2)

B. Tell whether the system of linear equations is consistent-independent, consistent-dependent, or inconsistent. Then
give the number of solution/s each system has.

1. 2. 3. 4.

1 3 15
𝑦 = 3𝑥 − 2 𝑦 = 𝑥+2 𝑦=− 𝑥− 𝑦 = 4𝑥 − 1
5. { 6. { 3 7. { 6 2 8. {
𝑦 = 3𝑥 + 1 𝑦 = −3𝑥 − 4 𝑦 = −0.5𝑥 − 7.5 𝑦 = −4𝑥 − 1

1 9
𝑦 = 4𝑥 + 5 𝑦 = 𝑥+3 5
𝑦 = 3𝑥 + 9 𝑦=0
9. { 9 10. { 9 11. { 12. {
𝑦 = 0.25𝑥 − 𝑦 =𝑥+3 𝑦=9 𝑥=0
5

4𝑥 − 8𝑦 + 3 = 0 1 1
2𝑥 + 3𝑦 = 5 3𝑥 − 5𝑦 = 12 𝑥
− 3 𝑦 = 10
13. { 14. { 15. { 3 16. { 2
2𝑥 − 3𝑦 = 5 9𝑥 − 15𝑦 = 0 𝑥 − 2𝑦 + 4 = 0 6𝑥 − 4𝑦 − 1 = 0

4 4 3 11 𝑦 − 3 = 3(𝑥 + 4)
𝑦 = −5𝑥 + 1 𝑦 = 3𝑥 𝑦 = −7𝑥 + 7
17. { 18. { 19. { 20. { 𝑥 𝑦
4𝑥 + 5𝑦 − 10 = 0 4𝑥 + 3𝑦 = 2 2𝑥 + 4𝑦 − 3 = 0 −5
+ 15 = 1

C. Find the solution to each system of linear equations by graphing.

𝑦 = 3𝑥 + 1 𝑦 = 4𝑥 + 3 𝑥+𝑦 =3 𝑦 =𝑥−3
1. { 2. { 3. { 4. {
𝑦 = −2𝑥 + 1 𝑦 = 4𝑥 − 2 2𝑥 − 𝑦 = 6 2𝑥 − 2𝑦 = 6

D. Find the solution to each system of linear equations by substitution.

1
𝑦 = −2𝑥 + 11 𝑦 = 2𝑥 − 7 𝑦 = −3𝑥 + 1 𝑥 − 5𝑦 = 1
1. { 2. { 3. { 4. {
𝑦 = 3𝑥 − 9 2𝑥 + 𝑦 − 5 = 0 𝑥 + 3𝑦 − 3 = 0 2𝑥 − 10𝑦 = −1

E. Find the solution to each system of linear equations by elimination.

4
𝑥 + 3𝑦 = 12 𝑥 + 5𝑦 = 3 3𝑥 − 2𝑦 = −3 𝑦 = −3𝑥 + 2
1. { 2. { 3. { 4.
3𝑥 − 3𝑦 = 4 3𝑥 + 15𝑦 = −2 4𝑥 − 5𝑦 = 3 2𝑥 − 𝑦 + 22 = 0

F. Translate the following mathematical statements words into symbols. Use the suggested variables.
1. The sum of two numbers 𝑥 and 𝑦 is 15 and their difference is 9.

2. The age of Leni (L) is half the age of Bernie (B). Five years ago, twice Leni’s age was 5 years less than Bernie’s.

3. An 𝑥 liters of 40% alcohol solution is mixed with y liters of 60% alcohol solution to produce 10 liters of 50% alcohol
solution.

4. Driving a car, Danny travelled from City A to City B at 𝑥 kph in 2 hours, and from City B to City C at 𝑦 kph in 2.5 hours.
He travelled a total distance of 30 km.

5. The sum of the digits 𝑥 and 𝑦 of a two-digit number is 10. If the digits are reversed the new number is 1 more than
twice the original number.
G. Solve each of the following problems.

1. The sum of two numbers is 24, and their difference is 8. What are the two numbers?

2. The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 36 more than the
original number. Find the original number.

3. You have a 10% salt solution and a 25% salt solution. How much of each solution should be mixed to obtain 15 liters
of a 20% salt solution?

4. A store sells two types of candy. One type costs P2 per piece, and the other costs P5 per piece. If you want to buy 10
pieces of candy and spend exactly P35, how many pieces of each type should you buy?

5. A boat travels 48 kilometers downstream in 4 hours and returns upstream in 6 hours. The speed of the current is
constant. Find the speed of the boat in still water and the speed of the current.

6. Two friends, Alice and Bob, are 120 kilometers apart. They start traveling towards each other at the same time. Alice
travels at 60 km/h, and Bob travels at 40 km/h. How long will it take for them to meet?

7. A car and a bus start from the same point and travel in opposite directions. The car travels 20 km/h faster than the
bus. After 3 hours, they are 300 kilometers apart. What are the speeds of the car and the bus?

8. A car left a station and traveled to Town A at 80 km/h. After 30 minutes, a second car left the same station and traveled
to Town A at 100 km/h. How long would it take the second car to catch up with the first car?

9. The sum of the ages of a father and his son is 50 years. Five years ago, the father was three times as old as his son.
How old are they now?

10. John invested a total of P15,000 in two accounts. One account pays 5% annual interest, and the other pays 7% annual
interest. At the end of the year, he earned a total of P930 in interest. How much did he invest in each account?

11. A rectangle's length is 3 meters longer than its width. The perimeter of the rectangle is 36 meters. What are the
dimensions of the rectangle?

12. In a school, the ratio of boys to girls is 3:5. If 10 boys are transferred to another school, the ratio of boys to girls
becomes 2:5. How many boys and girls are there in the school?

13. A plane travels from City A to City B in 2 hours with a tailwind and returns in 2.5 hours with a headwind. If the distance
between the cities is 400 kilometers, what are the speeds of the plane in still air and of the wind?

14. Two lines intersect to form a linear pair angles. One angle measures 20 degrees less than twice the measure of the
other angle. What are the measures of the two angles?

15. Alice can complete a task in 6 hours, while Bob can complete the same task in 4 hours. If they work together, how
long will it take them to complete the task?

Challenge: Solve each problem very carefully.

1. A factory machine can produce a batch of parts in 8 hours. If it runs alone for 3 hours and then is assisted by another
machine that can produce the batch in 5 hours, how long will it take them to complete the batch together?

2. The vertices of triangle ABC are 𝐴(3,5), 𝐵(1, −3) and 𝐶(7,1). Let 𝐷 be the midpoint of side 𝐵𝐶, and 𝐸 be the
midpoint of side 𝐴𝐶. At what point do lines 𝐴𝐷 and 𝐵𝐸 meet?

3. (PMO) Ella has three-fourths as much money as Jake. If Jake gives one-half of his money to Ella, and Ella gives Jake
one-fifth of what she has then, then Jake will have P1 less than Ella. How much had each at first?

Prepared by: Jose M. Manga Jr., Lagro High School, SDO-QC