For simultaneous equations containing fractions, decimals, or parentheses, first transform them into equations with integer coefficients, and then solve them.
When Parentheses Are Included
Example ①
{ 2(x+y)=10、3x−y=5
Solution: Expand the parentheses: 2x+2y=10 → x+y=5. The simultaneous equations are { x+y=5、3x−y=5. Add the equations: 4x=10→x=5/2,y=5/2.
{ 2(x+y)=10、3x−y=5
Solution: Expand the parentheses: 2x+2y=10 → x+y=5. The simultaneous equations are { x+y=5、3x−y=5. Add the equations: 4x=10→x=5/2,y=5/2.
When Decimals Are Included (Multiply by 10)
Example ②
{ 0.3x+0.2y=1.6、x+y=7
Solution: Equation 1 ×10: 3x+2y=16. { 3x+2y=16、x+y=7. Multiply ② by 2 and subtract: x=2,y=5。(2,5)
{ 0.3x+0.2y=1.6、x+y=7
Solution: Equation 1 ×10: 3x+2y=16. { 3x+2y=16、x+y=7. Multiply ② by 2 and subtract: x=2,y=5。(2,5)
When Fractions Are Included (Multiply by the LCM of the Denominators)
Example ③
{ x/2 + y/3 = 4、x−y=3
Solution: Equation 1 ×6: 3x+2y=24. { 3x+2y=24、x−y=3 → Add ②×2: 3x+2(x−3)=24... Using the elimination method, x=6,y=3。(6,3)
{ x/2 + y/3 = 4、x−y=3
Solution: Equation 1 ×6: 3x+2y=24. { 3x+2y=24、x−y=3 → Add ②×2: 3x+2(x−3)=24... Using the elimination method, x=6,y=3。(6,3)
Key Points
- Parentheses → Expand them first
- Decimals → Multiply by 10 or 100 to make integers
- Fractions → Multiply by the LCM of the denominators to make integers
- After transforming the equations, solve them using the elimination or substitution method
Practice Problems
- { 0.2x+0.3y=1.3、x+y=5 Solve the equation.
- { x/3+y/2=2、x−y=1 Solve the equation.
Answers and Explanations
- Answer:(x,y)=(2,3)。Equation 1 ×10→2x+3y=13. ②×2→2x+2y=10. Subtracting gives y=3,x=2。
- Answer:(x,y)=(3,2)。Equation 1 ×6→2x+3y=12. From ②, x=y+1→Substitute→2(y+1)+3y=12→y=2,x=3。