Solving Systems of Equations by Substitution

Solving Systems of Equations by Substitution

• Substitution is an algebraic method for finding exact solutions to systems of equations by replacing variables with equivalent expressions.

• Correct substitution eliminates the substituted variable, leaving an equation solvable for the other variable.

Steps for Solving

1. Solve one equation for one variable (e.g., y = 4x - 7).

2. Substitute the expression into the other equation (e.g., 4x - 7 = 2x + 5).

3. Solve for the remaining variable (e.g., x = 6).

4. Substitute the found value back into either equation to solve for the other variable (e.g., y = 4(6) - 7 = 17).

5. Check the solution in both original equations.

Rewriting Equations for Substitution

• If equations aren't in slope-intercept form, rearrange one to solve for x or y.

• Example: Given 2x + y = -10, solve for y to get y = -10 - 2x.

Special Cases

• Infinitely Many Solutions: If substitution results in a true statement (e.g., 10 = 10), the system has infinitely many solutions.

• No Solution: If substitution leads to a false statement (e.g., -4 = \frac{2}{5}), the system has no solution, indicating parallel lines.

Practice Problems

Easy

1. Solve the system:

   y = x + 1

   2x + y = 4

2. Solve the system:

   x = 2y

   x + y = 9

Medium

1. Solve the system:

   2x + y = 3

   3x - y = 7

2. Solve the system:

   4x + 2y = 10

   x - y = 1

Hard

1. Solve the system:

   \frac{1}{2}x + \frac{1}{3}y = 1

   x - y = 4

2. Solve the system:

   0.2x + 0.3y = 1.6

   0.5x - 0.1y = 1.1

Word Problems

Easy

1. The sum of two numbers is 20. The larger number is 4 more than the smaller number. Find the numbers.

2. A rectangle's length is twice its width. The perimeter is 36 cm. Find the length and width.

Medium

1. John invests $5000 in two accounts. One account pays 3% annual interest, and the other pays 4%. If his total interest for the year is $175, how much did he invest in each account?

2. A boat travels 24 miles downstream in 2 hours. The return trip against the current takes 3 hours. Find the speed of the boat in still water and the speed of the current.

Hard

1. A chemist has two acid solutions. One is 10% acid, and the other is 30% acid. How many liters of each solution should be mixed to obtain 50 liters of a 25% acid solution?

2. A person buys 2 pounds of coffee beans and 3 pounds of tea for $18. The next week, they buy 1 pound of coffee beans and 2 pounds of tea for $11. What is the price per pound of the coffee beans and tea