Chapter 1 Notes: Linear Graphs and Simultaneous Linear Equations

Horizontal and Vertical Lines

• Recall from general straight-line form: the equation of a straight line is in the form $y = mx + c$, where

  • $m$ is the gradient (slope) and

  • $c$ is the y-intercept.

• Horizontal lines

  • Gradient is zero: $m = 0$.

  • The line cuts the y-axis at $c$, so the equation is $y = c$.

  • All points on the line have the same y-coordinate (the line is horizontal).

• Vertical lines

  • Gradient is undefined (infinite slope).

  • All points on the line have the same x-coordinate, say $x = a$.

  • The equation is $x = a$.

Graphs of Linear Equations in the Form $ax + by = k$

• In unit 1.2, graphs are drawn for lines in the form $ax + by = k$, not only in slope-intercept form.

• LAPS Rule for drawing graphs (class investigation reference):

  • L – Labelling of the graph, equation, axes, origin, and parts of the question

  • A – Accuracy of the graph (must pass through the appropriate points and be drawn according to the equation)

  • P – Points plotted correctly with a cross

  • S – Scale used as given in the question

• Intercepts from $ax + by = k$:

  • x-intercept: set $y = 0$, solve for $x$ → $x = k/a$ (provided $a <br>eq 0$)

  • y-intercept: set $x = 0$, solve for $y$ → $y = k/b$ (provided $b <br>eq 0$)

• The relation between forms:

  • Converting between slope-intercept and standard form may require algebraic rearrangement.

Graphical Method for Solving Simultaneous Linear Equations

• When two lines are drawn corresponding to two linear equations in two variables, their point of intersection is the solution to the system (if it exists).

• Three possible situations:

  • One unique solution: the lines intersect at a single point (x, y).

  • Infinite solutions: the two lines coincide (they are the same line).

  • No solution: the lines are parallel but distinct.

• Important idea: a pair of linear equations ax + by = c and px + qy = r can be analyzed via the determinant Δ = $a q - b p$.

  • If $Δ <br>eq 0$, there is a unique solution, given by
    $x = \frac{c q - b r}{\Delta}, \quad y = \frac{a r - c p}{\Delta}.$

  • If $Δ = 0$, the lines are either coincident (infinite solutions) or parallel (no solution) depending on whether the equations are proportional.

• Graphical method steps (summary):

  • Draw both lines on the same grid.

  • Identify the intersection point (if any).

  • The intersection gives the solution pair $(x, y)$.

Solving Simultaneous Equations by Elimination Method

• Key idea: manipulate the equations so that one variable has the same absolute coefficient in both equations, allowing elimination.

• General steps:
  1) Label equations as (1) and (2).
  2) Choose which variable to eliminate (x or y) by comparing coefficients.
  3) Multiply one or both equations to obtain equal absolute coefficients for the chosen variable.
  4) Add or subtract the equations to eliminate that variable.
  5) Solve for the remaining variable.
  6) Substitute back to find the other variable and check in the original equations.

• Example outline (from transcript): Solve $\begin{cases} y + x = 6 \ y - x = \tfrac{2}{3} \end{cases}$

  • Multiply/add to eliminate one variable, then substitute to obtain the solution (the transcript shows an explicit worked flow and final result).

• Important notes:

  • If the coefficients of a variable are not the same in absolute value, multiply one (or both) equations to obtain a common multiple (LCM) for the chosen variable.

  • Always verify by substitution back into the original equations.

Solving Simultaneous Equations by Substitution Method

• Key idea: express one variable in terms of the other from one equation, then substitute into the other equation.

• General steps:
  1) Pick one equation and solve for one variable (often easier when its coefficient is 1).
  2) Substitute this expression into the other equation to obtain an equation in one variable.
  3) Solve for that variable.
  4) Substitute back to find the other variable.

• Notes from the transcript:

  • It is often advantageous to make the subject the variable that appears with a coefficient of 1 in one equation.

  • After obtaining a single-variable equation, back-substitute to get the second variable.

• Example outline (from transcript): Solve $\begin{cases} y + x = 6 \ y - x = 2/3 \end{cases}$

  • Solve the second equation for y (or x), substitute into the first, solve for one variable, then back-substitute to get the other variable.

Elimination vs Substitution: When to Use

• Elimination is often preferred when:

  • The coefficients of one variable can be easily matched by multiplying equations to obtain a common coefficient (especially using the LCM).

  • You want a straightforward elimination without dealing with fractions.

• Substitution is often preferred when:

  • One equation is already solved for a variable with coefficient 1 or easily isolated.

  • You want to substitute into the other equation after isolating a variable.

• Practical tips:

  • Always check your final answer by substituting into both equations.

  • For fractional coefficients, consider clearing fractions early by multiplying through.

Real-World Applications of Simultaneous Equations

• Four-step approach for word problems (Method 2 from transcript):
  1) Define the unknowns clearly, with units if any.
  2) Translate the problem into two equations in the unknowns.
  3) Solve the simultaneous equations by one of the algebraic methods.
  4) Reflect and conclude with a statement answering the original question.

• Example scenario: Cost of coffee and toast

  • Let the cost of one cup of coffee be $x$ dollars and one piece of toast be $y$ dollars.

  • Given two purchases, form two equations and solve for $x$ and $y$.

  • Use the solution to answer the question about real-world spending.

• The transcript also includes extensive worked examples and related problems (1C, 1D, etc.) that demonstrate:

  • Formulating equations from a scenario.

  • Solving with elimination, substitution, and occasionally graphical checks.

  • Interpreting the solutions in the context of the problem.

Summary of Key Concepts and Formulas

• Gradient and intercept:

  • Gradient (slope): $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$ for two points ((x1,y1), (x2,y2)).

  • y-intercept: the value of $y$ when $x = 0$.

• Horizontal line: $y = c$; gradient $m = 0$.

• Vertical line: $x = a$; gradient undefined.

• General line: $ax + by = k$ (standard form).

• Intercepts of $ax + by = k$:

  • x-intercept: $x = k/a\ (y=0)$

  • y-intercept: $y = k/b\ (x=0)$

• Two-equation systems:

  • Determinant method (for a system $ax + by = c$ and $px + qy = r$):

  • $\Delta = a q - b p$

  • If $\Delta \neq 0$, unique solution:
    $x = \frac{c q - b r}{\Delta}, \quad y = \frac{a r - c p}{\Delta}.$

  • If $\Delta = 0$, check if the equations are proportional to decide between coincident (infinite solutions) or parallel (no solution).

• Solving approaches:

  • Graphical method: find the intersection point of the two lines (if any).

  • Elimination method: eliminate one variable by equalizing coefficients and adding/subtracting equations.

  • Substitution method: express one variable in terms of the other and substitute.

• Types of solutions for two lines:

  • One solution: lines intersect at a single point.

  • Infinite solutions: lines coincide (identical).

  • No solution: lines are parallel but distinct.

Worked Problem Types to Practice (Based on Transcript)

• Graphing problems: draw lines given in either slope-intercept form or standard form; locate intersection.

• Elimination problems: manipulate coefficients to cancel one variable (pay attention to LCM when coefficients differ).

• Substitution problems: isolate a variable and substitute into the other equation.

• Worded problems: define unknowns, form two equations, solve, and state the conclusion in context.

• Fractional equations: often require clearing fractions before applying elimination or substitution.

Quick Reference: Important Formulas

• Gradient (slope): $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Horizontal line: $y = c$

• Vertical line: $x = a$

• General form: $ax + by = k$

• Intercepts from general form: x$-intercept = \frac{k}{a}, \quad y$-intercept = \frac{k}{b} (assuming nonzero a, b)

• Intersection solution (two-equation system):

  • For $ax + by = c$ and $px + qy = r$, with $\Delta = a q - b p$:

  • If $\Delta \neq 0$,
    $x = \frac{c q - b r}{\Delta}, \quad y = \frac{a r - c p}{\Delta}.$

• When solving by substitution or elimination, always check solutions in the original equations.