Practice: Linear relationships and simultaneous equations

Linear relationships and simultaneous equations

Warm-upHuman

Find the slope of the line through $(-2, 7)$ and $(4, -11)$ .

Linear relationships and simultaneous equations

Warm-upHuman

What is the slope of the line through $(2, -1)$ and $(2, 5)$ ?

Choose one

It is undefined: the line is vertical.

0

6

\tfrac{1}{6}

Linear relationships and simultaneous equations

CoreHuman

The line $y = -2x + b$ passes through the point $(3, 1)$ . What is $b$ ?

Linear relationships and simultaneous equations

CoreHuman

Which is the equation of the line through $(1, 4)$ and $(3, 10)$ ?

Choose one

y = 3x + 1

y = \tfrac{1}{3}x + \tfrac{11}{3}

y = 3x + 4

y = 6x - 2

Linear relationships and simultaneous equations

CoreHuman

Which of these relationships are proportional? (Choose all that apply.)

Choose all that apply

y = 4x

y = 4x + 1

The table

2 \mapsto 10

,

3 \mapsto 15

,

5 \mapsto 25

The table

1 \mapsto 3

,

2 \mapsto 5

,

3 \mapsto 7

y = x^2

Linear relationships and simultaneous equations

CoreHuman

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?

Linear relationships and simultaneous equations

CoreHuman

How many solutions does the system $2x - y = 3$ , $\;6x - 3y = 8$ have?

Choose one

NoneExactly oneExactly twoInfinitely many

Linear relationships and simultaneous equations

CoreFormal

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 $0$ .)

Choose one

∀ m b x₁ x₂ : Rat, x₁ ≠ x₂ → slope x₁ (m * x₁ + b) x₂ (m * x₂ + b) = m∀ m b x₁ x₂ : Rat, slope x₁ (m * x₁ + b) x₂ (m * x₂ + b) = m∃ x₁ x₂ : Rat, slope x₁ (2 * x₁ + 1) x₂ (2 * x₂ + 1) = 2∀ m b : Rat, slope 0 b 1 (m + b) = m

Linear relationships and simultaneous equations

StretchComputational

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.)

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Linear relationships and simultaneous equations

StretchHuman

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?