solving linear systems by substitution answer key

3 min read 23-11-2024

solving linear systems by substitution answer key

Meta Description: Master solving linear systems using the substitution method! This comprehensive guide provides clear explanations, step-by-step examples, and a handy answer key to practice problems. Perfect for students of all levels. Learn to solve systems of equations with confidence and improve your algebra skills.

Understanding Linear Systems

A linear system is a collection of two or more linear equations involving the same set of variables. The goal is to find the values of the variables that satisfy all equations simultaneously. This point represents the intersection of the lines represented by the equations (if they intersect).

One common method for solving linear systems is the substitution method. We'll explore this technique in detail, providing examples and a practice section with answers.

The Substitution Method: Step-by-Step

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, allowing you to solve for the remaining variable. Here's a breakdown:

Step 1: Solve for One Variable

Choose one of the equations and solve it for one of the variables. Select the equation and variable that makes this step easiest (often one with a coefficient of 1).

Step 2: Substitute

Substitute the expression you found in Step 1 into the other equation. This replaces the chosen variable with the equivalent expression.

Step 3: Solve the Equation

You now have a single equation with only one variable. Solve this equation using standard algebraic techniques.

Step 4: Substitute Back

Substitute the value you found in Step 3 back into either of the original equations (the one you solved for a variable or the other one). Solve for the remaining variable.

Step 5: Check Your Solution

Substitute both values (the x and y values) into both original equations. If both equations are true, your solution is correct.

Examples: Solving Linear Systems by Substitution

Let's work through a few examples to illustrate the substitution method.

Example 1:

Solve the system:

x + y = 5
x - y = 1

Solution:

Solve for one variable: From the first equation, we can easily solve for x: x = 5 - y
Substitute: Substitute this expression for x into the second equation: (5 - y) - y = 1
Solve: Simplify and solve for y: 5 - 2y = 1 => 2y = 4 => y = 2
Substitute back: Substitute y = 2 into x = 5 - y: x = 5 - 2 = 3
Check: 3 + 2 = 5 (True) and 3 - 2 = 1 (True). The solution is (3, 2).

Example 2:

Solve the system:

2x + y = 7
x - 3y = -10

Solution:

Solve for one variable: From the second equation, solving for x is easier: x = 3y - 10
Substitute: Substitute this into the first equation: 2(3y - 10) + y = 7
Solve: Simplify and solve for y: 6y - 20 + y = 7 => 7y = 27 => y = 27/7
Substitute back: Substitute y = 27/7 into x = 3y - 10: x = 3(27/7) - 10 = 81/7 - 70/7 = 11/7
Check: Substitute x = 11/7 and y = 27/7 into both original equations to verify.

Practice Problems with Answer Key

Now it's your turn! Try solving these systems using the substitution method.

Problem 1:

x + 2y = 7
x - y = 1

Problem 2:

3x + y = 11
x - 2y = -5

Problem 3:

2x - 4y = 6
x + y = 3

Answer Key:

Problem 1: (3, 2)

Problem 2: (3, 2)

Problem 3: (3, 0)

Special Cases: No Solution and Infinitely Many Solutions

Sometimes, you might encounter systems with no solution or infinitely many solutions.

No Solution: The resulting equation after substitution will be a false statement (e.g., 0 = 5). The lines represented by the equations are parallel and never intersect.
Infinitely Many Solutions: The resulting equation will be an identity (e.g., 0 = 0). The lines represented by the equations are the same line, and they overlap completely.

Conclusion

The substitution method is a powerful tool for solving linear systems. By following these steps and practicing regularly, you’ll develop confidence and proficiency in solving these types of problems. Remember to always check your solution! Mastering linear systems is a crucial step in your algebraic journey.