Simultaneous Equations Practice

What simultaneous equations actually mean

Two unknowns need two equations. A single equation such as x + y = 10 is true for an endless list of pairs, so it cannot settle what x and y are. A second, different equation cuts that list down to the one pair satisfying both at the same time, and that is what simultaneous means.

Elimination and substitution are the same idea in different clothes. Elimination adds or subtracts the two equations so that one variable cancels. Substitution rewrites one equation as y = something and drops that expression into the other. Both routes replace a pair of equations with one equation in one unknown, and both need the same final step: find the second value and check it in the equation you did not use. Geometrically, each linear equation describes a line, and a solution to the system is where the two lines intersect. One intersection gives one pair; parallel lines have no shared solution, while two equations describing the same line have many.

What this ladder covers

Elimination comes first on this ladder, and turning a sentence into a pair of equations comes last.

• What simultaneous means: explain why one equation cannot fix two values, and what the second one adds.
• Elimination by adding: line the equations up so that one variable cancels when they are added.
• Elimination by subtracting: deal with matching coefficients that carry the same sign.
• Substitution: use an equation already arranged as y = ... or x = ...
• Finding the partner value: substitute the first answer back, then check it in the other equation.
• From words to algebra: write "two numbers add to 20 and differ by 4" as a system and solve it.

Worked examples

Solve x + y = 10 and x − y = 2. Adding the equations takes out y: 2x = 12, so x = 6. Then y = 10 − 6 = 4. The +y and −y were built to cancel, which is why adding is the quick route.

Solve 2x + y = 11 and x − y = 1. Adding removes y again: 3x = 12, so x = 4, and y = 11 − 8 = 3. Check both: 2 × 4 + 3 = 11 and 4 − 3 = 1, so the pair fits each equation.

Solve y = x + 1 and 3x + y = 13. The first equation already names y, so substitute it: 3x + (x + 1) = 13. Then 4x + 1 = 13, 4x = 12, x = 3, and y = 3 + 1 = 4, which gives 9 + 4 = 13 in the second equation.

Two numbers add to 20 and differ by 4. Find them. Write x + y = 20 and x − y = 4. Adding gives 2x = 24, so x = 12 and y = 8. The smaller number is 8, because 12 − 8 = 4 and the two add to 20.

Solve 2x + 3y = 12 and 2x + y = 8. The x coefficients match, so subtract instead of adding: 2y = 4, y = 2. Putting that into the second gives 2x + 2 = 8, so x = 3. Matching coefficients with the same sign call for subtraction.

Mistakes worth watching for

Sign slips during elimination cost more answers than the method itself does.

• Adding the left sides but subtracting the right. 2x + y = 11 and x − y = 1 have to be handled the same way on both sides, which gives 3x = 12, not 3x = 10. A mixed operation invents an equation that was never asked.
• Finding x = 6 and stopping. Half an answer is not an answer. The question asks for a pair, so y has to be worked out and checked in the other equation.
• Subtracting when the signs already match. With +y in both equations, subtracting leaves 2y and never removes it. Adding is what cancels matching signs; subtracting cancels matching coefficients with opposite signs.
• Substituting y = x + 1 as y = x − 1. The sign of the constant is copied with the expression, not re-decided. Losing it changes 4x + 1 = 13 into 4x − 1 = 13 and the pair stops working.
• Checking one equation only. A pair can fit the equation used for the working and fail the other. The second equation is the one that catches sign errors.
• Forgetting what a solution represents. The answer is an ordered pair, such as (6, 4), not just x = 6. Both values together must make both original statements true.

How the ladder works

The ladder holds six rungs, a five-question Warm-Up first and sets of eight or nine later. A rung opens once 80% of first attempts are right, all six are available from the start, and every retry is a new draw.

Keep going

Make a printable set with the simultaneous equations worksheet generator, try the linear equation solver on a single unknown first, or read linear equations for the balance method this topic builds on.

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