The Lumenary · Grades 6–8

Linear relationships and simultaneous equations

Name: ______________________

Date: ______________________

You will learn to

Compute the slope of a line from two points and interpret it as a rate of change
Prove that the slope between any two points of the line $y = mx + b$ is $m$ , and that a constant rate of change gives a linear relationship
Write the equation of a line from its slope and a point, or from two points, and recognise linear and proportional relationships in tables, graphs and formulas
Solve two simultaneous linear equations by graphing, substitution and elimination
Explain why a system of linear equations has exactly one solution, no solution or infinitely many

1.
Find the slope of the line through $(-2, 7)$ and $(4, -11)$ .
2.
What is the slope of the line through $(2, -1)$ and $(2, 5)$ ?
- A.It is undefined: the line is vertical.
- B. $0$
- C. $6$
- D. $\tfrac{1}{6}$
3.
The line $y = -2x + b$ passes through the point $(3, 1)$ . What is $b$ ?
4.
Which is the equation of the line through $(1, 4)$ and $(3, 10)$ ?
- A. $y = 3x + 1$
- B. $y = \tfrac{1}{3}x + \tfrac{11}{3}$
- C. $y = 3x + 4$
- D. $y = 6x - 2$
5.
Which of these relationships are proportional? (Choose all that apply.)
- A. $y = 4x$
- B. $y = 4x + 1$
- C.The table $2 \mapsto 10$ , $3 \mapsto 15$ , $5 \mapsto 25$
- D.The table $1 \mapsto 3$ , $2 \mapsto 5$ , $3 \mapsto 7$
- E. $y = x^2$
6.
At a market in Nairobi, $3$ mangoes and $2$ avocados cost $160$ shillings, while $1$ mango and $2$ avocados cost $80$ shillings. How much does one mango cost, in shillings?
7.
How many solutions does the system $2x - y = 3$ , $\;6x - 3y = 8$ have?
- A.None
- B.Exactly one
- C.Exactly two
- D.Infinitely many
8.
Which Lean statement says "the slope between any two points of the line $y = mx + b$ is $m$ "? (Here slope x₁ y₁ x₂ y₂ is (y₂ - y₁) / (x₂ - x₁), and Lean defines division by zero to give $0$ .)
- A.∀ m b x₁ x₂ : Rat, x₁ ≠ x₂ → slope x₁ (m * x₁ + b) x₂ (m * x₂ + b) = m
- B.∀ m b x₁ x₂ : Rat, slope x₁ (m * x₁ + b) x₂ (m * x₂ + b) = m
- C.∃ x₁ x₂ : Rat, slope x₁ (2 * x₁ + 1) x₂ (2 * x₂ + 1) = 2
- D.∀ m b : Rat, slope 0 b 1 (m + b) = m
9.
Write fit_line(points) for a list of points (x, y) with distinct $x$ -values. If all the points lie on one line $y = mx + b$ , return the pair (m, b) as Fractions; otherwise return None. (Use exact fractions, not floats.)
Computational — Practice
10.
Let $f(x) = 3x + 2$ and $g(x) = 2x - 5$ . The function $x \mapsto g(f(x))$ (first $f$ , then $g$ ) is linear. What is its $y$ -intercept?

Answer key — Linear relationships and simultaneous equations

1.
$m = \dfrac{-11 - 7}{4 - (-2)} = \dfrac{-18}{6} = -3$ . The line falls $3$ units for every unit to the right.
2.
The run is $2 - 2 = 0$ , so $\frac{\text{rise}}{\text{run}} = \frac{6}{0}$ is undefined. Both points have $x = 2$ : they lie on the vertical line $x = 2$ , which has no slope.
3.
$1 = -2 \cdot 3 + b = -6 + b$ , so $b = 7$ . The line is $y = -2x + 7$ ; check: $-2 \cdot 3 + 7 = 1$ .
4.
$m = \frac{10 - 4}{3 - 1} = 3$ . From $4 = 3 \cdot 1 + b$ , $b = 1$ . So $y = 3x + 1$ . Check the second point: $3 \cdot 3 + 1 = 10$ .
5.
Proportional relationships are exactly the linear ones through the origin: (a) $y = 4x$ and (c) $y = 5x$ . (b) and (d) are linear with non-zero intercept; (e) is not linear.
6.
$(3m + 2a) - (m + 2a) = 160 - 80$ , so $2m = 80$ and $m = 40$ . Then $40 + 2a = 80$ gives $a = 20$ . Check: $3 \cdot 40 + 2 \cdot 20 = 160$ . A mango costs $40$ shillings.
7.
Multiplying the first equation by $3$ gives $6x - 3y = 9$ . A solution would make $6x - 3y$ equal to both $9$ and $8$ , which is impossible. In slope–intercept form the lines are $y = 2x - 3$ and $y = 2x - \frac{8}{3}$ : equal slopes, different intercepts — parallel.
8.
∀ m b x₁ x₂ : Rat, x₁ ≠ x₂ → slope x₁ (m * x₁ + b) x₂ (m * x₂ + b) = m. The hypothesis x₁ ≠ x₂ is essential: it makes the run non-zero, exactly as in the human proof, where we divide by $x_2 - x_1$ .

from fractions import Fraction

def fit_line(points):
    (x1, y1), (x2, y2) = points[0], points[1]
    m = Fraction(y2 - y1, x2 - x1)
    b = y1 - m * x1
    for x, y in points:
        if y != m * x + b:
            return None
    return m, b

Two points determine the line; the remaining points are then checked against it. Exact Fractions matter: with floats, $\frac{9}{5}$ would be stored as an approximation and the check y == m * x + b could fail by rounding.

10.
$g(f(x)) = 2(3x + 2) - 5 = 6x + 4 - 5 = 6x - 1$ . The slope is $2 \cdot 3 = 6$ and the intercept is $2 \cdot 2 + (-5) = -1$ , matching the general formula $m_1 m_2 x + (m_1 b_2 + b_1)$ with $m_1 = 2$ , $b_1 = -5$ , $m_2 = 3$ , $b_2 = 2$ .