Solving Simultaneous Equations

Introduction to Simultaneous Equations

Simultaneous equations are two or more equations that have the same variables. The goal is to find values for these variables that satisfy all the equations at the same time.

For example, consider the two linear equations:

• $2x + y = 7$
• $x - y = 1$

The solution is the pair $(x, y)$ that makes both equations true simultaneously.

Typically, simultaneous equations studied at IGCSE level involve two linear equations with two variables, such as $x$ and $y$.

Methods of Solving

There are three main methods to solve simultaneous equations:

• Substitution method: Solve one equation for one variable, then substitute into the other.
• Elimination method: Add or subtract the equations to eliminate one variable, then solve for the other.
• Graphical method: Draw both equations on a graph; the intersection point is the solution.

The graphical method is useful for visual understanding but less precise for exact answers. Substitution and elimination are algebraic methods that give exact solutions.

Solving by Substitution

The substitution method involves these steps:

1. Isolate one variable in one of the equations.
2. Substitute this expression into the other equation.
3. Solve the resulting single-variable equation.
4. Substitute back to find the other variable.

For instance, consider the system:

• $x + y = 5$
• $2x - y = 1$

Isolate $y$ from the first equation:

$$y = 5 - x$$

Substitute into the second equation:

$$2x - (5 - x) = 1$$

Simplify and solve for $x$:

$$2x - 5 + x = 1 \implies 3x = 6 \implies x = 2$$

Substitute $x = 2$ back to find $y$:

$$y = 5 - 2 = 3$$

So, the solution is $(x, y) = (2, 3)$.

Solving by Elimination

The elimination method involves:

1. Multiplying one or both equations if necessary to get coefficients of one variable to be the same (or opposites).
2. Adding or subtracting the equations to eliminate one variable.
3. Solving the resulting single-variable equation.
4. Substituting back to find the other variable.

For example, consider:

• $2x + 3y = 13$
• $3x - 3y = 3$

Add the two equations to eliminate $y$ because the coefficients $+3y$ and $-3y$ sum to zero:

$$(2x + 3y) + (3x - 3y) = 13 + 3 \implies 5x = 16 \implies x = \frac{16}{5} = 3.2$$

Substitute $x = 3.2$ into the first equation:

$$2(3.2) + 3y = 13 \implies 6.4 + 3y = 13 \implies 3y = 6.6 \implies y = 2.2$$

Solution: $(x, y) = (3.2, 2.2)$.

Checking Solutions

Always check your solution by substituting the values of $x$ and $y$ back into both original equations to verify they satisfy both.

For example, check $(x, y) = (2, 3)$ for the system:

• $x + y = 5$
• $2x - y = 1$

Substitute into the first:

$2 + 3 = 5$ ✓

Substitute into the second:

$2(2) - 3 = 4 - 3 = 1$ ✓

Since both are true, the solution is correct.

If substitution leads to a false statement (e.g. $5 = 2$), the system has no solution (the lines are parallel).

If substitution leads to a true statement with no variables (e.g. $0 = 0$), the system has infinitely many solutions (the equations represent the same line).

Graphically, no solution means the lines never intersect (parallel), and infinitely many solutions mean the lines coincide.