Finding the formula of a line

To write the formula $\displaystyle {y}={mx+b}$ of a line, you find two numbers: the slope $\displaystyle m$ and $\displaystyle b$.

• $\displaystyle m$ comes from the given slope, or from two points on the line.
• $\displaystyle b$ comes from a point on the line: it satisfies the formula, so you substitute it.

A line passes through the point $\displaystyle (2,5)$, and its slope is 3.
Find its formula.

The solution below substitutes the point in the formula and finds $\displaystyle b$:

• The slope is 3, so $\displaystyle {y}={3x+b}$.
• The point $\displaystyle (2,5)$ is on the line: we substitute $\displaystyle {x}={2}$ and $\displaystyle {y}={5}$, and get $\displaystyle {5}={3\cdot2+b}$.
• $\displaystyle {5}={6+b}$, so $\displaystyle {b}={-1}$.
• The formula: $\displaystyle {y}={3x-1}$.

A line passes through the points $\displaystyle (-1,7)$ and $\displaystyle (2,1)$.
Find its formula.

The solution below works out the slope first, and then finds $\displaystyle b$ from one of the points:

• $\displaystyle {m}={\frac{1-7}{2-(-1)}}={\frac{-6}{3}}={-2}$, so $\displaystyle {y}={-2x+b}$.
• We substitute $\displaystyle (2,1)$: $\displaystyle {1}={-2\cdot2+b}$, that is $\displaystyle {1}={-4+b}$, so $\displaystyle {b}={5}$.
• The formula: $\displaystyle {y}={-2x+5}$.

From the graph you read $\displaystyle b$ on the $\displaystyle y$-axis, and count $\displaystyle m$ in squares between two points the line passes through exactly:

• The line meets the $\displaystyle y$-axis at $\displaystyle (0,-1)$: $\displaystyle {b}={-1}$.
• From $\displaystyle (0,-1)$ to $\displaystyle (3,1)$: 3 right and 2 up, so $\displaystyle {m}={\frac{2}{3}}$.
• The formula: $\displaystyle {y}={\frac{2}{3}x-1}$.

Find the formula of the line parallel to $\displaystyle {y}={2x+3}$ through $\displaystyle P(3,1)$.

The solution below takes the slope from the given line, and finds $\displaystyle b$ from $\displaystyle P$:

• Parallel lines – the same slope: $\displaystyle {m}={2}$, so $\displaystyle {y}={2x+b}$.
• We substitute $\displaystyle P$: $\displaystyle {1}={2\cdot3+b}$, so $\displaystyle {b}={-5}$.
• The formula: $\displaystyle {y}={2x-5}$.

Find $\displaystyle b$ by solving an equation, and write the formula of each line.

Find the formula of the line through $\displaystyle A$ and $\displaystyle B$, and the formula of the line through $\displaystyle C$ and $\displaystyle D$.

Write the formula of each of the lines $\displaystyle f$, $\displaystyle g$ and $\displaystyle h$, from the graph.

Find the formula of the line parallel to $\displaystyle {y}={-x+3}$ through $\displaystyle P$.
Find the formula of the line parallel to $\displaystyle {y}={\frac{1}{3}x}$ through $\displaystyle Q$.

Are the three points on one line? Explain with slopes.

• $\displaystyle (0,1)$, $\displaystyle (2,5)$, $\displaystyle (5,11)$
• $\displaystyle (-1,-2)$, $\displaystyle (1,2)$, $\displaystyle (3,5)$

In a taxi the price grows at a constant rate with the number of kilometres.
A 4 km ride costs 26 ₪.
A 10 km ride costs 44 ₪.

• Write the formula of the price $\displaystyle y$ by the number of kilometres $\displaystyle x$.
• How much does getting into the taxi cost, before the first kilometre?
• How much will a 7 km ride cost?