Chapter 1 — Equations, functions, and basic graph interpretation¶

In Volume 1 we practiced algebraic manipulation. In this volume we use that toolkit to model relationships between variables. This chapter moves from expressions to first-degree equations and linear functions.

1.1 First-degree equations¶

An equation is an equality with an unknown, such as \(2x+3=11\). To solve it, we find the value of \(x\) that makes the equality true.

\[ 2x+3=11 \Rightarrow 2x=8 \Rightarrow x=4 \]

Worked example (aligned derivation)¶

Solve \(3(x-2)+4=2x+7\).

\[ \begin{aligned} 3(x-2)+4 &= 2x+7 \\ 3x-6+4 &= 2x+7 \\ 3x-2 &= 2x+7 \\ 3x-2x &= 7+2 \\ x &= 9 \end{aligned} \]

1.2 Linear function and parameter meaning¶

An affine (linear in this context) function can be written as

\[ f(x)=mx+b \]

where:

\(m\) is the slope of the line;
\(b\) is the vertical intercept, that is, the value when \(x=0\).

For \(f(x)=2x+1\):

\[ f(0)=1, \quad f(1)=3, \quad f(2)=5 \]

Value table:

\(x\)	\(f(x)=2x+1\)
0	1
1	3
2	5
3	7

1.3 Reading a line from two points¶

Even without plotting a full graph, two points determine a line. If the points are \((1,3)\) and \((3,7)\), then

\[ m=\frac{7-3}{3-1}=2 \]

Now use \(y=mx+b\) with \((1,3)\):

\[ 3=2\cdot 1+b \Rightarrow b=1 \]

So the function is \(y=2x+1\).

1.4 Pseudo-code to generate \((x, f(x))\) pairs¶

algorithm linear_function_table(m, b, x_start, x_end)
    for x from x_start to x_end
        y <- m*x + b
        print(x, y)
    end_for
end

This routine quickly produces data for numerical checks and graph reading.

1.5 Bridge to Volume 3¶

In Volume 3, sets of linear equations are rewritten with vectors and matrices. The scalar model \(mx+b\) is the entry point to systems with multiple unknowns.