Lesson 25 · Slope

Find the Missing Coordinate

If the slope is known, the missing point cannot hide.

A straight riding trail has one constant steepness. If you know that slope and most of two coordinates, you can reconstruct the missing value by asking what change in $y$ must match the change in $x$.

Δx = 4Δy = ?$(1,2)$$(5,y)$
Find ΔxUse Δy = m · ΔxAttach the startCheck the slope
Worked example

Slope $m=3$ through $(1,2)$ and $(5,y)$. First, $\Delta x=5-1=4$. Then $\Delta y=3\times4=12$. Since $y-2=12$, $y=14$. Check: $\frac{14-2}{5-1}=\frac{12}{4}=3$.

Guided practice
  1. Problem 1 · Missing y

    A line with slope $2$ passes through $(1,3)$ and $(5,y)$.

    1. Calculate $\Delta x$.
    2. Calculate the required $\Delta y$.
    3. Calculate $y$ and check the slope.
    Δx → slope × Δx → ending y → check →
  2. Problem 2 · Missing x

    A line with slope $-3$ passes through $(x,18)$ and $(7,6)$.

    1. Calculate $\Delta y=6-18$.
    2. Calculate the required $\Delta x$ using $\Delta x=\frac{\Delta y}{m}$.
    3. Calculate the missing $x$ and check.
    Δy → divide by slope → starting x → check →
  3. Problem 3 · From an intercept

    A line with slope $1.5$ passes through $(0,-2)$ and $(4,y)$.

    1. Calculate the required change in $y$.
    2. Calculate the missing coordinate $y$.
    m × Δx → add to starting y →
  4. Problem 4 · Riding lesson income

    Let $x=$ lessons taught and $y=$ total income in euros. The rate is €7 per lesson. The graph contains $(2,29)$ and $(6,y)$.

    1. Calculate $\Delta x$.
    2. Calculate $\Delta y$.
    3. Calculate $y$ with units.
    lesson change → income change → total →
  5. Problem 5 · Cooling

    Let $x=$ hours and $y=$ temperature in °C. A line has rate $-4$ °C/hour and passes through $(1,15)$ and $(x,3)$.

    1. Calculate $\Delta y$.
    2. Calculate $\Delta x$.
    3. Calculate the missing hour $x$.
    temperature change → time change → ending time →
  6. Problem 6 · Fractional slope

    A line has slope $\frac23$ and passes through $(3,5)$ and $(9,y)$.

    1. Calculate $\Delta x$.
    2. Calculate $\Delta y$.
    3. Calculate $y$.
    Δx → fraction × Δx → ending y →
Independent and retention checks
  1. Problem 7 · Missing x independently

    A line has slope $-\frac12$ and passes through $(-2,7)$ and $(x,3)$.

    1. Calculate the missing $x$.
    2. Check your result using $m=\frac{\Delta y}{\Delta x}$.
    write the slope equation yourself → solve → check →
  2. Problem 8 · Rate review

    A lake graph uses $x=$ days and $y=$ water level in centimetres. It passes through $(2,48)$ and $(6,36)$.

    1. Choose INCREASES or DECREASES.
    2. Calculate $\Delta y$.
    3. Calculate $\Delta x$.
    4. Calculate the slope with units.
    5. Interpret the rate in a sentence.
    direction → Δy → Δx → divide → meaning →
  3. Problem 9 · Inequality retention

    A phone has $30\%$ battery and loses $4$ percentage points each hour. Let $h=$ hours.

    1. Write an inequality for when the battery is below $10\%$.
    2. Solve the inequality and interpret the result.
    model → isolate negative term → reverse sign →
  4. Problem 10 · Geometric retention

    A photo sequence has sizes $243,81,27,9,\ldots$ KB.

    1. Calculate $r$.
    2. Write an explicit formula.
    3. Calculate $a_6$.
    ratio → formula → substitute →
Show answers
Problem 1a
$\Delta x=5-1=4$.
Problem 1b
$\Delta y=m\Delta x=2\times4=8$.
Problem 1c
$y=3+8=11$; $\frac{11-3}{5-1}=2$.
Problem 2a
$\Delta y=6-18=-12$.
Problem 2b
$\Delta x=\frac{-12}{-3}=4$.
Problem 2c
$7-x=4$, so $x=3$; $\frac{6-18}{7-3}=-3$.
Problem 3a
$\Delta y=1.5(4-0)=6$.
Problem 3b
$y=-2+6=4$.
Problem 4a
$\Delta x=6-2=4$ lessons.
Problem 4b
$\Delta y=€7\times4=€28$.
Problem 4c
$y=€29+€28=€57$.
Problem 5a
$\Delta y=3-15=-12$ °C.
Problem 5b
$\Delta x=\frac{-12}{-4}=3$ hours.
Problem 5c
$x=1+3=4$ hours.
Problem 6a
$\Delta x=9-3=6$.
Problem 6b
$\Delta y=\frac23\times6=4$.
Problem 6c
$y=5+4=9$.
Problem 7a
$\Delta y=3-7=-4$ and $\Delta x=\frac{-4}{-1/2}=8$, so $x=-2+8=6$.
Problem 7b
$\frac{3-7}{6-(-2)}=\frac{-4}{8}=-\frac12$.
Problem 8a
DECREASES.
Problem 8b
$\Delta y=36-48=-12$ cm.
Problem 8c
$\Delta x=6-2=4$ days.
Problem 8d
$m=-12\div4=-3$ cm/day.
Problem 8e
The lake level decreases by $3$ centimetres per day.
Problem 9a
$30-4h<10$.
Problem 9b
$$30-4h<10\quad\mid-30$$$$-4h<-20\quad\mid\div(-4)$$$$h>5$$The battery is below $10\%$ after more than $5$ hours.
Problem 10a
$r=\frac13$.
Problem 10b
$a_n=243(\frac13)^{n-1}$.
Problem 10c
$a_6=1$ KB.
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