How to Solve Simultaneous Equations Step by Step

Suppose you have x plus y equals 10 and x minus y equals 2. Add the equations together and the y terms cancel, leaving 2x equals 12. That gives x equals 6. Substitute 6 into either original equation, such as x plus y equals 10, and you get y equals 4. The solution is the pair x equals 6 and y equals 4 because both equations are true for those values.

Elimination works best when one variable already has matching or opposite coefficients. If the coefficients do not match, multiply one or both equations first. For example, if one equation contains 2x and another contains 3x, you could multiply to create 6x in both. Then add or subtract the equations so one variable disappears. Keep every term on both sides of an equation multiplied by the same factor.

A common mistake in solving simultaneous equations is losing a negative sign during subtraction. If you subtract an entire equation, every term in that equation changes sign. Writing the equations neatly underneath each other reduces this risk. After one variable is found, substitute it back carefully and solve for the second. Do not stop after finding only one value, because the solution is a coordinate pair.

Substitution is useful when one equation already gives a variable on its own, such as y equals 2x plus 1. If the second equation is x plus y equals 10, replace y with 2x plus 1. You then have x plus 2x plus 1 equals 10, which simplifies to 3x equals 9. So x equals 3, and substituting back gives y equals 7.

This method explains how to solve simultaneous equations as a replacement process. Since y and 2x plus 1 are equal, one can stand in for the other. The result is an equation with only x. You can also rearrange an equation first to make substitution possible. If 2x plus y equals 8, then y equals 8 minus 2x, which can be inserted into the other equation.

Always check your final pair in both original equations. If x equals 3 and y equals 7, place those numbers into each equation and confirm that both left sides equal the correct right sides. This catches arithmetic slips and sign errors quickly. Graphically, the same solution is the point where two straight lines cross, which gives a useful visual meaning to the algebra. Some pairs of equations behave differently. Parallel lines have no shared solution, while two equations describing the same line have infinitely many solutions. Algebra reveals these cases when variables cancel and leave an impossible statement or an identity. Recognising these outcomes is part of solving simultaneous equations, because not every pair produces one neat coordinate.

Many simultaneous equation questions begin as words. Suppose a cinema sells 200 tickets, adult tickets cost 8 pounds and child tickets cost 5 pounds, and the takings are 1,360 pounds. Let a be adult tickets and c child tickets. The count gives a plus c equals 200, and the money gives 8a plus 5c equals 1,360. Substituting c equals 200 minus a gives 3a plus 1,000 equals 1,360, so a equals 120 and c equals 80.

Substitution also handles cases where one equation is not a straight line. Take y equals x squared and y equals x plus 2. Setting them equal gives x squared equals x plus 2, which rearranges to x squared minus x minus 2 equals 0. This factorises to x minus 2 times x plus 1, so x equals 2 or x equals minus 1. The matching y values are 4 and 1, giving two solutions, the two points where the line crosses the curve.

Solving several equations at once is far older than modern algebra. The Chinese text known as the Nine Chapters on the Mathematical Art, compiled roughly two thousand years ago, sets out a method using counting rods laid out in columns on a board. One famous problem concerns three grades of grain. Three bundles of top grade, two of middle grade and one of low grade yield 39 units. Two, three and one bundles yield 34. One, two and three bundles yield 26. The method removes unknowns step by step, much like modern elimination.

You can solve it the same way. Call the yields per bundle x, y and z. Subtract the second equation from the first to get x minus y equals 5. Multiply the first equation by 3 and subtract the third, which removes z and leaves 8x plus 4y equals 91. Substituting x equals y plus 5 gives 12y plus 40 equals 91, so y equals 4.25 and x equals 9.25. Putting these into the first equation gives z equals 2.75. With three unknowns, the key is to remove the same variable twice, leaving two equations in two unknowns.

Balancing a chemical equation is really a set of simultaneous equations. Take propane burning in oxygen: a C3H8 plus b O2 gives c CO2 plus d H2O, where a, b, c and d are the unknown numbers of each molecule. Each element must balance. Carbon gives 3a equals c, hydrogen gives 8a equals 2d, and oxygen gives 2b equals 2c plus d. There are four unknowns but only three equations, because doubling every number would still balance, so you choose a equals 1. Then c equals 3, d equals 4 and 2b equals 10, so b equals 5.

Satellite navigation uses the same idea on a larger scale. Your phone does not know its three position coordinates, and its clock is not perfectly synchronised with the very precise clocks on the satellites, which adds a fourth unknown. Each satellite signal gives one equation linking the receiver's position and clock error to its distance from that satellite. With four unknowns, the receiver needs signals from at least four satellites, and it solves the resulting system many times a second. Because these equations involve distances, they are not straight lines, so the receiver solves them by repeated approximation.

For two equations written as ax plus by equals e and cx plus dy equals f, there is a ready-made formula. First calculate the number ad minus bc, called the determinant. Then x equals ed minus bf, divided by the determinant, and y equals af minus ce, divided by it. Try 2x plus 3y equals 12 and 4x minus y equals 10. The determinant is 2 times minus 1, minus 3 times 4, which gives minus 14. For x, the top is minus 12 minus 30, or minus 42, so x equals 3. For y, the top is 20 minus 48, or minus 28, so y equals 2.

This method, called Cramer's rule after the Swiss mathematician Gabriel Cramer, who published a general version in 1750, has a useful warning built in. If the determinant is zero, you cannot divide by it, and that tells you the lines are parallel or identical before you do any other work. Take 2x plus 3y equals 6 alongside 4x plus 6y equals 12. The determinant is 12 minus 12, which is zero, and the second equation is just the first one doubled, so there are infinitely many solutions. Change the 12 to 15 and the lines become parallel, with no solution at all.

Solving simultaneous equations comes down to removing one unknown, solving for the other, and then substituting back. Use elimination when coefficients line up neatly and substitution when one variable is already isolated or easy to isolate. Keep signs organised and check the final pair in both equations. The correct answer must satisfy both at the same time.