Chapter 2 — Systems of equations and comparison of linear functions¶

In Chapter 1 we modeled one variable as a function of another with expressions such as \(f(x)=mx+b\). Now we move to situations where two linear relationships act at the same time. This naturally leads to systems of equations and graphical interpretation through intersections.

2.1 A \(2\times 2\) linear system as the meeting of two lines¶

A two-unknown linear system can be written as

\[ \begin{cases} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2 \end{cases} \]

Each equation represents a line in the Cartesian plane. The system solution is the point \((x,y)\) that satisfies both equations simultaneously.

If we isolate \(y\), we get affine forms:

\[ y=m_1x+n_1 \quad \text{and} \quad y=m_2x+n_2 \]

Comparing slopes and intercepts helps predict system behavior.

2.2 Graph interpretation: three scenarios¶

For two lines, there are three classic cases:

Relationship between lines	Typical condition	System interpretation
Intersecting	\(m_1\neq m_2\)	One solution (unique intersection)
Distinct parallel	\(m_1=m_2\) and \(n_1\neq n_2\)	No solution
Coincident	\(m_1=m_2\) and \(n_1=n_2\)	Infinitely many solutions

This view connects algebra and geometry: solving the system means finding the intersection.

2.3 Worked example: intersection point¶

Consider

\[ f(x)=2x+1 \quad \text{and} \quad g(x)=-x+7 \]

To find where they intersect, set the outputs equal:

\[ 2x+1=-x+7 \]

Then

\[ 3x=6 \Rightarrow x=2 \]

Substitute back:

\[ y=f(2)=2\cdot 2+1=5 \]

So the intersection point is

\[ (2,5) \]

Equivalent system form:

\[ \begin{cases} y=2x+1 \\ y=-x+7 \end{cases} \]

2.4 Comparing growth of linear functions¶

When comparing affine functions, slope \(m\) measures the rate of change. If \(m_f>m_g\), then \(f\) grows faster than \(g\).

Example:

\[ f(x)=3x-2, \quad g(x)=x+4 \]

Difference:

\[ f(x)-g(x)=(3x-2)-(x+4)=2x-6 \]

To know when \(f\) exceeds \(g\):

\[ 2x-6>0 \Rightarrow x>3 \]

for \(x<3\), \(f(x)<g(x)\);
for \(x=3\), \(f(x)=g(x)\);
for \(x>3\), \(f(x)>g(x)\).

2.5 Introductory matrix form for a system¶

Even in Volume 2, we can organize a system using matrix notation. For

\[ \begin{cases} 2x+y=5 \\ -x+y=1 \end{cases} \]

we write

\[ \begin{bmatrix} 2 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix} \]

This structure prepares the transition to Volume 3, where vector and matrix operations are developed more systematically.

2.6 Structured comparison: substitution vs elimination¶

Method	Core idea	Good use case
Substitution	Isolate one variable and replace in the other equation	One equation is already easy to isolate
Elimination	Combine equations to cancel one variable	Coefficients make cancellation straightforward

Both methods should produce the same solution set.

2.7 Chapter wrap-up¶

This chapter consolidates the idea that linear systems describe the interaction of linear relationships. Graphical intersection and algebraic solving are complementary views of the same model, and matrix notation opens the path to the linear algebra tools of Volume 3.