Lesson 33 · Graphs of Two-Variable Equations

Plot the Art-Shop Plan

Turn an art shop's price rule into a picture you can read at a glance.

Papelaria Azul, Lisbon

Mia walks into a small art shop in Lisbon. A sign says: every sketchbook costs €3, and the carry bag costs €4. Mia wants to know the cost of any number of sketchbooks without asking at the till. An equation does that with symbols, and a graph does it with a picture.

In this lesson you will build a table of values (a list of $x$ and $y$ pairs), plot each ordered pair $(x,y)$ as a dot, join the dots into a line, choose which graph matches an equation, and read what the intercepts mean. An intercept is a place where the line meets an axis.

Worked example · From equation to table to graph

The shop's rule is $y=3x+4$, where $x$ is the number of sketchbooks bought and $y$ is the total cost in euros.

Step 1 · Table. Substitute each chosen $x$ with parentheses, for example $y=3(2)+4=10$.

$x$ (sketchbooks)	0	1	2	3	4	5
$y$ (euros)	4	7	10	13	16	19

Step 2 · Ordered pairs. Write each column as $(x,y)$, always $x$ first.

(0, 4)(1, 7)(2, 10)(3, 13)(4, 16)(5, 19)

Step 3 · Plot and draw. Go across to $x$, then up to $y$, and mark a dot. The dots line up, so draw one straight line through them.

Step 4 · Read the highlights. The red dot $(0,4)$ is the $y$-intercept, where the line crosses the $y$-axis. It is the cost of $0$ sketchbooks: the €4 bag. The green dot $(3,13)$ says $3$ sketchbooks cost €13. Every step right by $1$ goes up by $3$, the price of one sketchbook.

Check: $3(3)+4=13$, so $(3,13)$ is on the graph and satisfies the equation. A shop sells only whole sketchbooks, so strictly only the dots are real prices; the line shows the pattern.

Choosing a graph: a three-question test

1. Where does the line start on the $y$-axis? Substitute $x=0$. 2. Does the line go up or down as $x$ grows? The sign of the number multiplying $x$ tells you. 3. Does a second point, such as $x=1$, land on the line?

Practice · Table, plot, line

Scaffolded: the plan is shown

1 · Pencils at the Lisbon shop. Loose pencils cost €2 each and the bag costs €3, so $y=2x+3$, where $x$ is the number of pencils and $y$ is the total cost in euros.
(a) Determine the three coordinates for $x=0$, $2$, and $4$, showing one substitution as evidence.
(b) Interpret the first and last coordinates in context.

substitute x-values → write coordinates → interpret the start and one later point

2 · Plot the pencil prices. Use your three pairs from Problem 1 on this grid (print it or copy it on squared paper). The $x$-axis is pencils and the $y$-axis is euros.

(a) Plot the three coordinates from Problem 1 and draw one straight line through them.
(b) Predict the cost of $3$ pencils from your line, then verify it by substituting $x=3$ into the equation.

plot → ruler line → read → substitute to verify

3 · Paint money running down. Mia has €20 to spend on paint tubes at €2 each. Her money left is $y=20-2x$, where $x$ is the number of tubes bought and $y$ is euros left.
(a) Calculate $y$ for $x=0$, $x=5$ and $x=10$, and write the three pairs.

(b) Plot the three pairs and draw the line. Does it go up or down?
(c) Name the point where your line meets the $x$-axis and explain what it means for Mia's money. Frame: "After buying $\_\_$ tubes, Mia has €$\_\_$ left."

substitute → dots → line slopes down because of −2 → where does y reach 0?

From here, test and build graphs yourself.

Choose the matching graph

4 · Brush kit. A Kyoto-style brush kit costs €2 per brush plus a €4 carry tube: $y=2x+4$, where $x$ is the number of brushes and $y$ is the total cost in euros. Three graphs are shown (the scale is the same on each: $x$ from 0 to 5, $y$ from 0 to 14).

Graph A

Graph B

Graph C

(a) Calculate $y$ at $x=0$ and at $x=1$, and write the two pairs.
(b) Choose the graph (A, B or C) that matches the equation.
(c) Explain what mistake one of the other graphs shows. Frame: "Graph $\_$ is wrong because its line $\_\_\_$."

x = 0 gives the start → second point → direction up/down → eliminate

5 · Watercolour credit. Mia has a €6 shop credit. Each pan of watercolour uses €2 of it, so $y=6-2x$, where $x$ is the number of pans and $y$ is the credit left in euros. (Same scale on each graph: $x$ from 0 to 6, $y$ from 0 to 14.)

Graph A

Graph B

Graph C

(a) Choose the graph that matches $y=6-2x$.
(b) Explain how the sign of $-2$ and the starting value $6$ helped you rule out the other two graphs.

start value at x = 0 → minus sign means falling → check where it reaches 0

Build and read a graph with a negative start

6 · Mia's sketch-fund balance. Mia buys a €6 brush on credit. Then she sells drawings for €3 each. Her balance is $y=3x-6$, where $x$ is drawings sold and $y$ is her balance in euros ($y$ below $0$ means she owes money).
(a) Determine the coordinates for $x=0$, $2$, and $4$, showing enough working to verify them.

(d) Plot your three pairs and draw the line.
(e) Name the $y$-intercept and explain what it means for Mia.
(f) Name the $x$-intercept and explain what it means for Mia.

negative start → dots below the axis → line crosses x-axis → intercepts have meaning

Spaced review

7 · Ink budget. An ink budget graph contains $(1,14)$ and $(5,6)$, where $x$ is days and $y$ is euros of ink budget left.
Calculate and interpret the signed slope in euros per day, showing the coordinate changes you used.

end minus start for both → divide → sign says falling → units

8 · Sketchbook stack. A graph of total cost passes through $(2,5)$ and has slope $3$ euros per sketchbook. Another point on it has $x=6$.
Determine the missing coordinate and show how the slope connects the two points.

Δx → use slope to find Δy → add to the known y

9 · Is the price rate constant? A table of sketchbook orders: $x=1$, $3$, $6$ sketchbooks cost $y=7$, $13$, $22$ euros.
(a) Calculate the rate from $x=1$ to $x=3$.
(b) Calculate the rate from $x=3$ to $x=6$.
(c) Decide whether the rate is constant and explain why.

uneven x-steps → divide each interval → compare the rates

Retention check

10 · Canvas and frame. A canvas is $40$ cm by $30$ cm. A frame goes around the edge and paint covers the whole face.
(a) Calculate the frame length.
(b) Calculate the painted area, with units.

frame traces the boundary → paint covers the surface → units

Show answers

Problem 1a · x = 0

$y=2(0)+3=3$, so $(0,3)$.

Problem 1a · x = 2

$y=2(2)+3=7$, so $(2,7)$.

Problem 1a · x = 4

$y=2(4)+3=11$, so $(4,11)$.

Problem 1b · first coordinate

$(0,3)$ means $0$ pencils cost €3 (the bag fee).

Problem 1b · last coordinate

$(4,11)$ means $4$ pencils cost €11. Each extra pencil adds €2.

Problem 2a

All three dots lie on one straight line starting at $(0,3)$ on the $y$-axis and rising to $(4,11)$.

Problem 2b

Read up from $x=3$ to the line: $y=9$. Check: $y=2(3)+3=9$, so $3$ pencils cost €9. The line and the equation agree.

Problem 3a

$y=20-2(0)=20$; $y=20-2(5)=10$; $y=20-2(10)=0$. Pairs: $(0,20)$, $(5,10)$, $(10,0)$.

Problem 3b

The line slopes down from $(0,20)$ to $(10,0)$.

Problem 3c

$(10,0)$ is the $x$-intercept (where the line meets the $x$-axis): after buying $10$ tubes Mia has €0 left, so the whole €20 is spent.

Problem 4a

$y=2(0)+4=4$, so $(0,4)$. $y=2(1)+4=6$, so $(1,6)$.

Problem 4b

Graph C: it starts at $4$ on the $y$-axis and rises $2$ for every $1$ step right.

Problem 4c

Graph A is $y=-2x+4$ (wrong sign: it falls, but the cost must rise). Graph B is $y=4x+2$ (start and rate swapped: it starts at €2, not €4, and climbs at €4 per brush). Either explanation is correct.

Problem 5a

Graph B: $y=-2x+6$ starts at $6$ on the $y$-axis and falls to $0$ at $x=3$. Check: $y=6-2(3)=0$.

Problem 5b

The rate is $-2$: each pan used takes €2 away, so the line goes down. Graph A rises (wrong sign) and Graph C starts at $3$ instead of $6$ (it used the $x$-intercept $3$ as the start).

Problem 6a · x = 0

$y=3(0)-6=-6$, so $(0,-6)$.

Problem 6a · x = 2

$y=3(2)-6=0$, so $(2,0)$.

Problem 6a · x = 4

$y=3(4)-6=6$, so $(4,6)$.

Problem 6b

Plot $(0,-6)$, $(2,0)$, $(4,6)$ and draw one straight line rising to the right.

Problem 6c

$y$-intercept $(0,-6)$: before selling anything Mia's balance is $-$€6, she owes €6 for the brush.

Problem 6d

$x$-intercept $(2,0)$: after selling $2$ drawings her balance is €0, the break-even point.

Problem 7 · vertical change

$\Delta y=6-14=-8$ euros.

Problem 7 · horizontal change

$\Delta x=5-1=4$ days.

Problem 7 · slope and meaning

$m=\frac{-8}{4}=-2$ euros per day (the ink budget falls by €2 each day).

Problem 8 · coordinate change

$\Delta y=m\cdot\Delta x=3\times(6-2)=3\times4=12$.

Problem 8 · missing coordinate

$y=5+12=17$, so the point is $(6,17)$.

Problem 9a

From $x=1$ to $3$: $\frac{13-7}{3-1}=\frac{6}{2}=3$ euros per sketchbook.

Problem 9b

From $x=3$ to $6$: $\frac{22-13}{6-3}=\frac{9}{3}=3$ euros per sketchbook.

Problem 9c

Yes, the rate is constant: both intervals give $3$, so the points lie on one straight line (it is the worked example's $y=3x+4$).

Problem 10a

$P=2(40+30)=140$ cm of frame.

Problem 10b

$A=40\times30=1200$ cm$^2$ of canvas.