Simultaneous Linear Equations (2 Variables)

Simultaneous Linear Equations (2 Variables)#

Linear Equations Revision#

Theory#

Before solving systems of linear equations, let’s review single linear equations. A linear equation in one variable has the form:

\[ax + b = 0\]

where \(a \neq 0\). The solution is \(x = -\frac{b}{a}\).

For two variables, a linear equation takes the form:

\[ax + by = c\]

This represents a straight line in the coordinate plane. A single linear equation in two variables has infinitely many solutions - all the points on the line.

Application#

Examples#

Example 1#

Find three solutions to: \(2x + 3y = 12\)

Solution:

\(2x + 3y = 12 \quad \text{(Choose values for x and solve for y)}\)

When \(x = 0\): \(3y = 12\), so \(y = 4 \quad \text{(Solution: (0, 4))}\)

When \(x = 3\): \(6 + 3y = 12\), so \(y = 2 \quad \text{(Solution: (3, 2))}\)

When \(x = 6\): \(12 + 3y = 12\), so \(y = 0 \quad \text{(Solution: (6, 0))}\)

Interactive Visualization: Linear Equations in Two Variables#

Multiple Choice Questions#

Simultaneous Linear Equations (2 Variables)#

Theory#

A system of simultaneous linear equations consists of two or more linear equations that must be satisfied at the same time. For two equations in two variables:

\[\begin{split}\begin{align} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{align}\end{split}\]

The solution is the point \((x, y)\) where both lines intersect. There are three possible outcomes:

  1. Unique Solution: Lines intersect at exactly one point

  2. No Solution: Lines are parallel (inconsistent system)

  3. Infinitely Many Solutions: Lines are identical (dependent system)

The main methods for solving simultaneous linear equations are:

  1. Graphical Method: Plot both lines and find their intersection

  2. Substitution Method: Solve one equation for a variable and substitute

  3. Elimination Method: Add or subtract equations to eliminate a variable

  4. Matrix Method: Use matrices and row operations

Tip

Always check your solution by substituting the values back into both original equations. This verifies correctness and catches arithmetic errors.

Interactive Visualization: Simultaneous Linear Equations#

Application#

Examples#

Example 1#

Solve by substitution: \(\begin{cases} x + 2y = 7 \\ 3x - y = 4 \end{cases}\)

Method 1: Substitution

\(x + 2y = 7 \quad \text{(Solve for x in terms of y)}\)

\(x = 7 - 2y \quad \text{(Substitute into second equation)}\)

\(3(7 - 2y) - y = 4 \quad \text{(Expand and simplify)}\)

\(21 - 6y - y = 4 \quad \text{(Combine like terms)}\)

\(21 - 7y = 4 \quad \text{(Solve for y)}\)

\(-7y = -17\), so \(y = \frac{17}{7} \quad \text{(Find x)}\)

\(x = 7 - 2(\frac{17}{7}) = 7 - \frac{34}{7} = \frac{15}{7} \quad \text{(Solution: \)(\frac{15}{7}, \frac{17}{7})\()}\)

Example 2#

Solve by elimination: \(\begin{cases} 2x + 3y = 11 \\ 5x - 2y = 4 \end{cases}\)

Method 1: Elimination

\(\begin{cases} 2x + 3y = 11 \\ 5x - 2y = 4 \end{cases} \quad \text{(Multiply first by 2, second by 3)}\)

\(\begin{cases} 4x + 6y = 22 \\ 15x - 6y = 12 \end{cases} \quad \text{(Add equations to eliminate y)}\)

\(19x = 34 \quad \text{(Solve for x)}\)

\(x = \frac{34}{19} = \frac{34}{19} \quad \text{(Substitute back)}\)

\(2(\frac{34}{19}) + 3y = 11 \quad \text{(Solve for y)}\)

\(\frac{68}{19} + 3y = 11\), so \(3y = \frac{209 - 68}{19} = \frac{141}{19}\)

\(y = \frac{47}{19} \quad \text{(Solution: \)(\frac{34}{19}, \frac{47}{19})\()}\)

Example 3#

Identify the nature of the system: \(\begin{cases} 2x + 4y = 6 \\ x + 2y = 3 \end{cases}\)

Method 1: Analysis

\(2x + 4y = 6 \quad \text{(Divide by 2)}\)

\(x + 2y = 3 \quad \text{(Compare with second equation)}\)

\(x + 2y = 3 \quad \text{(Both equations are identical)}\)

Therefore, the system has infinitely many solutions. All points on the line \(x + 2y = 3\) are solutions.

Multiple Choice Questions#

Sector Specific Questions: Simultaneous Linear Equations Applications#

Key Takeaways#

Important

  1. A system of two linear equations in two variables can have one solution, no solution, or infinitely many solutions

  2. Graphically, the solution is where the two lines intersect

  3. Substitution method: solve one equation for a variable and substitute into the other

  4. Elimination method: add or subtract equations to eliminate a variable

  5. For unique solution: lines intersect at one point (different slopes)

  6. For no solution: lines are parallel (same slope, different y-intercepts)

  7. For infinitely many solutions: lines are identical (same slope and y-intercept)

  8. Always verify solutions by substituting back into both original equations

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