Unraveling Linear Simultaneous Equations

 

What are Linear Simultaneous Equations?

Linear simultaneous equations are two or more equations that share the same variables (e.g., $x$ and $y$) 

and are solved together to find the values of the variables.

Think of it like a puzzle where the equations must work together to reveal the solution. For example: $$2x + y = 7 \quad \text{and} \quad x - y = 1$$

Here, both equations need to be true at the same time

 

Methods to Solve Linear Simultaneous Equations

There are three main methods:

1. Substitution
2. Elimination
3. Graphical Method (not covered in this section)

We’ll focus on substitution and elimination, as they are the most commonly used.

 

Quick Tip: Always verify your solutions by substituting them back into the original equations

 

Method 1: Substitution

In substitution, you solve one equation for one variable and substitute it into the other equation.

Example 1

Solve the simultaneous equations: $$x + y = 5 \\  2x - y = 4$$

1. Step 1: Solve one equation for one variable (pick the simpler equation). From $x + y = 5$: $$y = 5 - x$$
2. Step 2: Substitute into the second equation. Replace $y$ in $2x - y = 4$: $$2x - (5 - x) = 4$$
3. Step 3: Simplify and solve for $x$: $$2x - 5 + x = 4 \\ 3x - 5 = 4 \\ 3x = 9 \\ x = 3$$
4. Step 4: Substitute $x = 3$ back into the first equation: $$x + y = 5 \\  3 + y = 5 \\ y = 2$$

Solution: $$x = 3, \, y = 2$$

 

 

 

 

 

 

Method 2: Elimination

In elimination, you add or subtract the equations to eliminate one variable.

Example 2

Solve the simultaneous equations: $$3x + 2y = 12 \\ 2x - 2y = 4$$

1. Step 1: Add or subtract the equations to eliminate one variable: Add the equations together to cancel out $y$: $$(3x + 2y) + (2x - 2y) = 12 + 4 \\ 5x = 16 \\ x = \frac{16}{5}$$
2. Step 2: Substitute $x = \frac{16}{5}$ into one of the original equations: Use $3x + 2y = 12$: $$3 \left(\frac{16}{5}\right) + 2y = 12 \\ \text{Simplify: }  \\ \frac{48}{5} + 2y = 12 \\ \text{Multiply through by 5 to eliminate fractions: } \\  48 + 10y = 60 \\ 10y = 12 \\ y = \frac{6}{5}$$

Solution: $$x = \frac{16}{5}, \, y = \frac{6}{5}$$

 

 

 

 

 

Method 3: Graphical Method

What Does Solving Graphically Mean? Solving simultaneous linear equations graphically means drawing both equations as straight lines on a graph. The point where the two lines meet (intersect) is the solution to the equations—it gives the values of $x$ and $y$ that work for both equations.

Example 3

Solve the following equations graphically $$y = 2x + 1 \\ y =-x + 4$$

Steps to Solve Graphically

• Step 1: Rewrite both equations in the form $y = mx + c$ if they’re not already. This is the slope-intercept form where $m$ is the slope and $c$ is the $y$-intercept
• Step 2: Plot the first equation $y = 2x + 1$
  • Start at $c = 1$ on the $y$-axis
  • Use the slope $m = 2$ Move up 2 units and right 1 unit to plot the next point.
  • Draw the line through these points.
• Step 3: Plot the second equation $y = -x + 4$:
  • Start at $c = 4$ on the $y$-axis
  • Use the slope $m = 1$: Move down 1 unit and right 1 unit to plot the next point.
  • Draw the line through these points.
• Step 4: Identify the intersection point of the two lines. This point is the solution to the simultaneous equations

Solution From the graph, the two lines intersect at $x = 1, y = 3$. So the solution is $$x = 1, y = 3$$