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Build: Equations, functions and straight lines

Algebra · step 3 of 5

A question to start

Two phone plans. Plan A costs 30 dollars a month plus 2 dollars per gigabyte. Plan B costs 10 dollars a month plus 4 dollars per gigabyte. Plan B is cheaper if you hardly use data; Plan A is cheaper if you use a lot. Where exactly is the switch-over point? On this page you will answer it three ways: with an equation, with a graph, and with a pair of equations solved together.

The idea

Equations with the letter on both sides

The two plans cost the same when \(30 + 2g = 10 + 4g\). Now the unknown is on both sides. The plan is always the same:

  1. Expand brackets, if there are any.
  2. Collect the letter terms on one side and the plain numbers on the other, doing the same to both sides.
  3. Divide by the coefficient to get the letter alone.
  4. Check by substituting.
\[ \begin{aligned} 30 + 2g &= 10 + 4g\\ 30 &= 10 + 2g \qquad \text{(take } 2g \text{ from both sides)}\\ 20 &= 2g \qquad \text{(take 10 from both sides)}\\ g &= 10 \end{aligned} \]

At 10 gigabytes both plans cost \(30 + 20 = 50\) dollars, and \(10 + 40 = 50\) dollars. Any equation like this, where the letter appears only to the first power (no squares), is called linear.

Functions: a machine with one output

A function is a rule that takes an input and gives exactly one output. Think of a machine: a number goes in, the rule acts, a number comes out. We name the function, often \(f\), and write \(f(x)\) (read "f of x") for the output when the input is \(x\).

\[f(x) = 2x + 30 \qquad \Rightarrow \qquad f(10) = 2 \times 10 + 30 = 50.\]

The brackets in \(f(10)\) do not mean multiply; they mean "put 10 into \(f\)". The set of inputs a function allows is its domain; the set of outputs it can give is its range. For the phone plan, the domain is the numbers of gigabytes from 0 upwards.

A real example: Mars's gravity pulls on things only about 0.38 times as hard as Earth's. So if \(W\) is an object's weight on Earth, its weight on Mars is the function \(M(W) = 0.38\,W\). A suitcase that weighs 200 newtons on Earth (the newton is the unit of force) weighs \(M(200) = 76\) newtons on Mars.

Graphs and coordinates

A function's graph is a picture of all its input–output pairs. Each pair \((x, y)\) is a point: go \(x\) across and \(y\) up from the origin \((0, 0)\). For a linear function, the points lie on a straight line.

Plan APlan B 1050 3010 gigabytesdollars

The lines cross at \((10, 50)\): the switch-over point. Left of it, Plan B's line is lower (cheaper); right of it, Plan A's is.

Slope and intercept: \(y = mx + b\)

Every non-vertical straight line can be written as

\[y = mx + b\]
  • \(b\) is the y-intercept: where the line crosses the vertical axis, the value of \(y\) when \(x = 0\). For Plan A, \(b = 30\), the fixed monthly fee.
  • \(m\) is the slope: how much \(y\) goes up when \(x\) goes up by 1. For Plan A, \(m = 2\), the cost per gigabyte. A bigger slope means a steeper line; a negative slope means the line goes down from left to right.

The slope between any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line is "rise over run":

\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]

Two lines with the same slope are parallel: they never meet.

Two equations at once

Finding where two lines cross means finding one pair \((x, y)\) that makes both equations true. This is a system of equations (also called simultaneous equations). Two good methods:

  • Substitution. Get one letter alone in one equation, and put that expression into the other.
  • Elimination. Add or subtract the equations (or multiples of them) so that one letter cancels.

For the phone plans, \(y = 30 + 2g\) and \(y = 10 + 4g\). Substituting the first into the second gives \(30 + 2g = 10 + 4g\), the very equation we solved at the top. Graphs, single equations and systems are three views of the same idea.

Worked examples

1. Brackets on both sides. Solve \(3(x + 4) = 5x - 2\).

  • Expand: \(3x + 12 = 5x - 2\).
  • Take \(3x\) from both sides: \(12 = 2x - 2\).
  • Add 2 to both sides: \(14 = 2x\), so \(x = 7\).
  • Check: left \(3(7 + 4) = 33\); right \(5 \times 7 - 2 = 33\). Equal.

2. The line through two points. Find the equation of the line through \((1, 3)\) and \((4, 9)\).

  • Slope: \(m = \dfrac{9 - 3}{4 - 1} = \dfrac{6}{3} = 2\).
  • So \(y = 2x + b\). Put in the point \((1, 3)\): \(3 = 2 + b\), so \(b = 1\).
  • The line is \(y = 2x + 1\). Check with the other point: \(2 \times 4 + 1 = 9\). Correct.

3. Elimination. Solve \(2x + y = 11\) and \(x - y = 1\) together.

  • Add the two equations. The \(y\) terms cancel: \(3x = 12\), so \(x = 4\).
  • Put \(x = 4\) into \(x - y = 1\): \(4 - y = 1\), so \(y = 3\).
  • Check in the first: \(2 \times 4 + 3 = 11\). The lines cross at \((4, 3)\).

Try it

  1. Solve \(5(x - 2) = 2x + 11\).
  2. For \(f(x) = 3x - 4\), find \(f(5)\), and find the input \(x\) that gives \(f(x) = 20\).
  3. Find the equation of the line through \((-2, 1)\) and \((2, 9)\).
  4. Solve the system \(x + y = 10\) and \(x - y = 4\).
  5. Using \(M(W) = 0.38\,W\), what does an object weigh on Earth if it weighs 30.4 newtons on Mars?
  6. Where do the lines \(y = 3x + 2\) and \(6x - 2y = 8\) meet?
Answers
  1. Expand: \(5x - 10 = 2x + 11\). Take \(2x\) from both sides: \(3x - 10 = 11\). Add 10: \(3x = 21\), so \(x = 7\). Check: \(5 \times 5 = 25\) and \(14 + 11 = 25\).
  2. \(f(5) = 15 - 4 = 11\). For \(3x - 4 = 20\): \(3x = 24\), so \(x = 8\).
  3. \(m = \dfrac{9 - 1}{2 - (-2)} = \dfrac{8}{4} = 2\). Then \(1 = 2 \times (-2) + b\) gives \(b = 5\), so \(y = 2x + 5\). Check: \(2 \times 2 + 5 = 9\).
  4. Add the equations: \(2x = 14\), so \(x = 7\); then \(y = 10 - 7 = 3\). Check: \(7 - 3 = 4\).
  5. \(0.38\,W = 30.4\), so \(W = 30.4 \div 0.38 = 80\) newtons.
  6. Nowhere. Rearranging the second: \(2y = 6x - 8\), so \(y = 3x - 4\). Both lines have slope 3, so they are parallel, and since their intercepts differ (2 and −4) they never meet. The system has no solution.

Common mistake

Getting the sign wrong when expanding a bracket with a minus in front. \(-(x - 3)\) is \(-x + 3\), not \(-x - 3\): the minus multiplies every term inside. In \(10 - 2(x - 3)\), write \(10 - 2x + 6 = 16 - 2x\). If in doubt, put in a number: with \(x = 5\), the original is \(10 - 2 \times 2 = 6\), and \(16 - 10 = 6\) agrees, while \(10 - 2x - 6 = -6\) does not.

Where it leads