Solving Systems of Linear Equations: Substitution & Elimination

Solving Systems of Linear Equations: Substitution & Elimination

Below are step‑by‑step instructions for both methods, assuming a system of two equations in two variables (e.g., ax+by=cax+by=c and dx+ey=fdx+ey=f).

Substitution Method

Idea: Solve one equation for one variable, then substitute that expression into the other equation.

Steps:

1. Isolate a variable – Choose one equation and solve for one variable in terms of the other.
   Example: From x+2y=5x+2y=5, get x=5−2yx=5−2y.

2. Substitute – Replace that variable in the other equation with the expression you found.
   Example: If the second equation is 3x−y=13x−y=1, replace xx with 5−2y5−2y:
   3(5−2y)−y=13(5−2y)−y=1.

3. Solve for the remaining variable – Simplify and solve the resulting one‑variable equation.
   Example: 15−6y−y=115−6y−y=1 → 15−7y=115−7y=1 → −7y=−14−7y=−14 → y=2y=2.

4. Back‑substitute – Plug the found value into the expression from step 1 to get the other variable.
   Example: x=5−2(2)=1x=5−2(2)=1.

5. Check (optional) – Verify the solution in both original equations.

Elimination Method (also called Addition/Subtraction)

Idea: Combine the equations so that one variable cancels out.

Steps:

1. Align the equations – Write both equations in standard form Ax+By=CAx+By=C.

2. Create opposite coefficients – Choose a variable to eliminate. Multiply one or both equations by suitable numbers so that the coefficients of that variable are opposites (e.g., 33 and −3−3).

3. Add (or subtract) the equations – The chosen variable cancels, leaving a single equation in the other variable.

4. Solve for the remaining variable – Find its value.

5. Substitute back – Plug that value into either original equation to find the second variable.

6. Check (optional) – Confirm the solution satisfies both equations.

Example:

{2x+3y=74x−3y=5{2x+3y=74x−3y=5​

Add the equations → 6x=126x=12 → x=2x=2.
Substitute into first equation: 2(2)+3y=72(2)+3y=7 → 4+3y=74+3y=7 → 3y=33y=3 → y=1y=1.
Solution: (2,1)(2,1).

When to use which?

• Substitution is often easier when one variable already has a coefficient of 11 or −1−1.

• Elimination is convenient when coefficients are small or already opposites.

Both methods work for any linear system; choose the one that looks simpler for your equations.

To solve the word problem step-by-step, let’s break it down into the two parts: (a) determining the population in 2018, and (b) finding out when the population was about 30,000.

Given:

• Population equation: $P = 14t^2 + 820t + 42000$

• Time variable (t): the number of years since 2008, so for example:   - When $t = 0$, it's the year 2008.   - When $t = 10$, it's the year 2018.

a) Determine the population in 2018:

1. Calculate t for the year 2018:    - Since 2018 - 2008 = 10, we have $t = 10$.

2. Substitute t into the population equation:    - $P = 14(10)^2 + 820(10) + 42000$
      - $P = 14(100) + 8200 + 42000$    - $P = 1400 + 8200 + 42000$    - $P = 51400$

The population in 2018 is 51,400.

b) When was the population about 30,000?

1. Set the population equation to 30,000:    - $30000 = 14t^2 + 820t + 42000$

2. Rearrange the equation:    - $14t^2 + 820t + 42000 - 30000 = 0$    - $14t^2 + 820t + 12000 = 0$

3. Simplify (divide all terms by 14):    - $t^2 + 58.57t + 857.14 = 0$

4. Use the quadratic formula to solve for t:    - The quadratic formula is: $t = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$    - Here, $a = 1, b = 58.57, c = 857.14$    - Calculate the discriminant:      - $D = b^2 - 4ac = (58.57)^2 - 4(1)(857.14)$      - $D = 3433.68 - 3428.56 = 5.12$

5. Calculate the roots:    - $t = rac{-58.57 \pm \sqrt{5.12}}{2(1)}$    - Compute $an(5.12)$, which is approximately $2.26$    - Therefore,      - $t_1 = rac{-58.57 + 2.26}{2} o t ext{ is invalid (negative)}$      - $t_2 = rac{-58.57 - 2.26}{2} o t ext{ is a valid solution}$    - Calculate:
        - $t_2 = rac{-60.83}{2} = -30.415$ (not realistic here, try modifying using a calculated discriminant)

Final Assessment:

• After stepwise consideration, the valid positive assessment indicates other modifiers would yield an approximate boarders for populations under 30,000 but seem structural without positive analysis in respective boundaries of individual outputs, thus on population ageing.

1. Read the Problem Carefully
   
      - Begin by reading the problem thoroughly to understand what is being asked.
   
      - Identify the important information and details related to the problem.

2. Identify the Variables
   
      - Determine which quantities you need to find (unknowns).
   
      - Define variables to represent these unknowns clearly.

3. Set Up Equations
      - Based on your translations, write down the equations needed to solve the problem.
      - Make sure the equations accurately reflect the relationships described in the problem.

4. Solve the Equations
      - Use appropriate mathematical methods to solve for the unknowns (e.g., algebraic manipulation, substitution, or graphing).
      - If applicable, check if you have one or multiple solutions.

5. Interpret the Solution
      - Once you have the solution(s), substitute back into the context of the problem to ensure they make sense.
      - Answer the question posed in the problem clearly and concisely.

6. Check Your Work
      - Review your steps and calculations to ensure there are no errors.
      - Verify that your solution meets the criteria set by the problem.

7. Write a Complete Answer
      - Present your final answer in a clear, complete sentence that addresses the question.
      - Include units if applicable.