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Unit 2: Systems of Linear Equations and Inequalities

Topic 1

Solving Systems by Substitution

A system of linear equations is a set of two or more linear equations with the same variables. A solution to the system is an ordered pair (x,y)(x, y) that makes all equations true at the same time.
{y=2x+33x−y=9\begin{cases} y = 2x + 3 \\ 3x - y = 9 \end{cases}
The substitution method involves solving one equation for one variable and substituting that expression into the other equation.
Steps:
  1. Solve one equation for one variable.
  2. Substitute that expression into the other equation.
  3. Solve for the remaining variable.
  4. Substitute back to find the other variable.
  5. Check the solution in both equations.

Core Skills

  • Choose the easier equation to isolate a variable.
  • Substitute carefully and use parentheses when replacing variables.
  • Verify the final ordered pair in both equations.

Example 1: Solving by Substitution

Solve the system:
{y=2x+1x+y=10\begin{cases} y = 2x + 1 \\ x + y = 10 \end{cases}
Step 1: Substitute y=2x+1y = 2x + 1 into the second equation.
x+(2x+1)=10x + (2x + 1) = 10
Step 2: Simplify and solve for xx.
3x+1=10⇒3x=9⇒x=33x + 1 = 10 \quad \Rightarrow \quad 3x = 9 \quad \Rightarrow \quad x = 3
Step 3: Substitute x=3x = 3 back into y=2x+1y = 2x + 1.
y=2(3)+1=7y = 2(3) + 1 = 7
Final Answer: (3,7)\boxed{(3, 7)}

Example 2: Substitution with Both Variables on One Side

Solve the system:
{2x+y=11x=2y−4\begin{cases} 2x + y = 11 \\ x = 2y - 4 \end{cases}
Step 1: Substitute x=2y−4x = 2y - 4 into the first equation.
2(2y−4)+y=112(2y - 4) + y = 11
Step 2: Simplify.
4y−8+y=11⇒5y=19⇒y=195=3.84y - 8 + y = 11 \quad \Rightarrow \quad 5y = 19 \quad \Rightarrow \quad y = \dfrac{19}{5} = 3.8
Step 3: Substitute y=3.8y = 3.8 back into x=2y−4x = 2y - 4.
x=2(3.8)−4=7.6−4=3.6x = 2(3.8) - 4 = 7.6 - 4 = 3.6
Final Answer: (3.6,3.8)\boxed{(3.6, 3.8)}

Key Takeaways

  • Always isolate one variable first.
  • Substitute carefully and simplify completely.
  • Write your solution as an ordered pair (x,y)(x, y).