Systems: Solving by Addition or Subtraction

Systems of equations

A linear equation in one unknown has but one root: the value of the unknown is a constant. A linear equation in two unknowns, like x + y = 6, is satisfied by an unlimited number of pairs of values. Any change in one is accompanied by a change in the other, so the unknowns are really variables, as they were called in the chapter on graphs.

• Two or more equations in two or more variables form a system of equations.
• A system satisfied by the same values of the variables is a simultaneous system.
• A set of values, one for each variable, that satisfies an equation or a system is a set of roots.

An equation with one root or a limited number of roots is determinate: 2x = 10 is satisfied by x = 5 only, and n^2 - 5n = -6 by n = 2 or 3 only. In the same way the system \begin{cases} x + 2y = 12 \\ 4x - y = 3 \end{cases} is a determinate system: only x = 2, y = 5 satisfies it. (The system x^2 + y^2 = 25, x + y = 7 is determinate too; it has just two sets of roots, x = 4, y = 3 and x = 3, y = 4.)

An equation or system satisfied by an unlimited number of sets of roots is indeterminate. x + y + z = 10 is satisfied by 2, 3, 5 and by 4, 5, 1 and many more; the system x + y + z = 8, 2x - 3y - z = 9 is satisfied by x = 7, y = 2, z = -1 and by x = 5, y = -1, z = 4, among many others.

Solution by addition or subtraction

Example 1. Solve \begin{cases} x + 4y = 4 & (1) \\ x - 2y = 16 & (2) \end{cases}

Eliminate x first. Subtracting, (1) - (2) gives 6y = -12, so y = -2. Substituting -2 for y in (1), x - 8 = 4 and x = 12.

Check: 12 - 8 = 4 and 12 + 4 = 16. ✓

Example 2. Solve \begin{cases} 13x + 3y = 14 & (1) \\ 7x - 2y = 22 & (2) \end{cases}

Eliminate y first:

Substituting 2 for x in (2), 14 - 2y = 22, so y = -4. Check: 26 - 12 = 14 and 14 + 8 = 22. ✓

Either letter could have been eliminated first. To eliminate x the multipliers would be 7 and 13; for y they are the handier 2 and 3. The notation (3) - (4) means subtract the members of (4) from those of (3); (3)\cdot 6 means multiply both members of (3) by 6. So we speak of adding or subtracting two equations and of multiplying or dividing an equation by a number.

Rule. If necessary, multiply the first equation by one number and the second by another, so that the coefficients of the same variable in both are numerically equal. If these coefficients have like signs, subtract one equation from the other; if unlike signs, add. Solve the equation obtained. Substitute the value found in the simplest equation containing both variables and solve for the other.

Check. Substitute the values in the original equations and simplify until you get obvious identities.

When the method breaks down

Try the rule on 3x - 6y = 40 and x - 2y = 8. Multiplying the second by 3 gives 3x - 6y = 24; subtracting, 0 = 16, an impossibility. No pair of values satisfies both: the system is incompatible or inconsistent. Its graphs are parallel lines.

Try it on x + 2y = 8 and 3x + 6y = 24: you get 0 = 0. The second equation is just the first multiplied by 3, so every set of roots of one is a set of roots of the other. If we regard them as different equations, the system has an infinite (unlimited) number of sets of roots: its graphs are the same line.

How to type your answers

Type each value in its own box: a whole number, a fraction like -17/5, or a mixed number like 4 1/2.

• If the system is inconsistent (no set of roots), type none in every box.
• If it has an unlimited number of sets of roots, type any in the box for the first letter and give the other letter(s) in terms of it.

For an unlimited system like x + 2y = 8, 3x + 6y = 24: type any for x and y = \frac{8 - x}{2} for y.

Try one: