A pair of two equations containing the two variables x and y is called a system of simultaneous equations. A pair of values for x and y that satisfies both equations at the same time is a solution.
Checking a Solution
Substitute the values found into both equations. If both equations hold, they are a solution. Checking only one equation is not sufficient.
Example ①
Check whether (x, y)=(3, 2) is a solution of the simultaneous equations { x+y=5、2x−y=4.
Answer:Equation 1: 3+2=5 ✓、equation 2: 2×3−2=4 ✓. Both hold, so it is a solution.
Check whether (x, y)=(3, 2) is a solution of the simultaneous equations { x+y=5、2x−y=4.
Answer:Equation 1: 3+2=5 ✓、equation 2: 2×3−2=4 ✓. Both hold, so it is a solution.
Example ②
Find all pairs of natural numbers (x, y) satisfying x + 2y = 8.
Answer:y=1→x=6、y=2→x=4、y=3→x=2. There are three pairs: (6,1)、(4,2)、(2,3)。
Find all pairs of natural numbers (x, y) satisfying x + 2y = 8.
Answer:y=1→x=6、y=2→x=4、y=3→x=2. There are three pairs: (6,1)、(4,2)、(2,3)。
Key Points
- Simultaneous equations: find (x,y) satisfying both equations at the same time
- Always substitute into both equations when checking a solution
- A single linear equation in two variables has infinitely many solutions
Practice Problems
- Check whether (x,y)=(1,4) is a solution of { 2x+y=6、x−y=−3.
- Find all pairs of natural numbers (x,y) satisfying 2x+y=7.
Answers and Explanations
- Answer:Equation 1: 2+4=6 ✓、equation 2: 1−4=−3 ✓. It is a solution.
- Answer:(1,5)、(2,3)、(3,1)。
Explanation:y=7−2x, and y is a natural number when x=1,2,3。