Simultaneous Equations and Substitution for the TMUA

Understanding Equations and Their Graphs

An equation like $y = x + 2$ is essentially a concise way to represent an infinite set of number pairs $(x, y)$. For example, $(1, 3)$, $(2, 4)$, and $(\pi, \pi + 2)$ are all members of this set. Geometrically, we represent these pairs on a Cartesian plane as the line $y = x + 2$. Every point on that line corresponds to a pair in the set. Moving between the algebraic set of pairs and the geometric picture of the graph is a fundamental skill for university admission tests.

When we talk about simultaneous equations, we are looking for the number pairs that appear in both sets at the same time. Geometrically, this corresponds to the points where the graphs of the equations cross.

Solving Linear Simultaneous Equations

Consider the following two linear equations:

1. $x + 2y = 5$
2. $2x + y = 4$

To solve these simultaneously, we can use three primary methods: substitution, elimination, or graphical analysis.

By Substitution

We rearrange one equation to make one variable the subject and then substitute this into the other equation. In equation (1), we can write $x = 5 - 2y$. Substituting this into equation (2) gives:

$2(5 - 2y) + y = 4$

$10 - 4y + y = 4$

$10 - 3y = 4$

$3y = 6 \Rightarrow y = 2$

Since $x = 5 - 2y$, then $x = 5 - 2(2) = 1$. The solution is $(1, 2)$.

By Elimination

Alternatively, we can manipulate the equations to make the coefficients of one variable identical. Multiplying equation (1) by 2 gives $2x + 4y = 10$. We then subtract equation (2) from this new equation:

$(2x + 4y) - (2x + y) = 10 - 4$

$3y = 6 \Rightarrow y = 2$

Substituting $y = 2$ back into $x + 2y = 5$ yields $x = 1$.

Graphical Interpretation

Drawing both lines on the $xy$ plane reveals they intersect at a single point, $(1, 2)$.

For any pair of linear equations, there are three geometric possibilities:

1. Parallel and distinct lines: These have the same gradient but different intercepts. They never cross, meaning there are no solutions. For example, $y + 2x = 4$ and $y + 2x = 8$.
2. Parallel and identical lines: These are the same line written in different ways. Every point on the line is a solution, meaning there are infinitely many solutions. For example, $y + 2x = 4$ and $2y + 4x = 8$.
3. Non-parallel lines: These lines have different gradients and must cross exactly once. These systems always have one unique solution.

Linear and Quadratic Simultaneous Equations

Things become more interesting when one equation is linear and the other is a quadratic. A quadratic equation generally takes the form $y = ax^2 + bx + c$. When solving these simultaneously with a line, we usually use substitution to eliminate one variable, resulting in a single quadratic equation in terms of $x$ or $y$.

There are three geometric possibilities for the intersection of a line and a quadratic:

1. The line crosses the quadratic: There are two distinct real solutions.

1. The line is tangent to the quadratic: The line just touches the curve at one point. This results in one repeated real solution.

1. The line never crosses the quadratic: There are no real solutions.

Worked Example: Linear and Quadratic

Solve $y = x^2 + 3x + 2$ and $y = x + 1$ simultaneously.

To solve, we eliminate $y$ by setting the two expressions for $y$ equal to each other:

$x^2 + 3x + 2 = x + 1$

Rearranging into the standard quadratic form $ax^2 + bx + c = 0$:

$x^2 + 2x + 1 = 0$

This factorises to:

$(x + 1)^2 = 0$

This gives a single repeated solution for $x$:

$x = -1$

To find the $y$ coordinate, we substitute $x = -1$ into the simpler linear equation:

$y = (-1) + 1 = 0$

Because there is only one repeated solution at $(-1, 0)$, we know geometrically that the line $y = x + 1$ is a tangent to the parabola $y = x^2 + 3x + 2$.

Further Exploration

While the TMUA focuses on linear and quadratic systems, you should consider the implications for higher-order polynomials. For example, a quadratic and a cubic could have up to three intersection points. Unlike a line and a quadratic, a line and a cubic must cross at least once. This is because the ends of a cubic function go to positive infinity in one direction and negative infinity in the other, eventually forcing an intersection with any line.