Simultaneous Equations Formula Explained with Examples

The Standard Form You Must Know

Every pair of linear simultaneous equations can be written in one neat shape. Using plain letters, it looks like this:

a x + b y = c and d x + e y = f

Here x and y are the unknowns. The letters a, b, d and e are the coefficients, which are the numbers beside the unknowns. The letters c and f are the constants on the right side.

NCERT Class 10 writes the same pair as a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0. The two forms mean the same thing. Only the position of the constant changes, so take care with signs when you move between them.

The Main Formulas at a Glance

Method	Formula or rule
Cramer’s rule, determinant D	D = ae – bd
Determinant for x	Dx = ce – bf
Determinant for y	Dy = af – cd
Solution	x = Dx / D and y = Dy / D, when D is not zero
Substitution	x = (c – b y) / a, then put into the second equation
Elimination	Multiply to match coefficients, then add or subtract

All these routes lead to the same answer. Cramer’s rule is a direct formula, which is why most calculators and computer programs use it.

Where Cramer’s Formula Comes From

You do not have to memorise blindly. The formula comes straight from elimination.

1. Multiply the first equation by e: a e x + b e y = c e.
2. Multiply the second equation by b: b d x + b e y = b f.
3. Subtract the second from the first. The y terms cancel: (ae – bd) x = ce – bf.
4. Divide both sides by (ae – bd). You get x = (ce – bf) / (ae – bd).

The same steps, with d and a as multipliers, give y = (af – cd) / (ae – bd). So Cramer’s rule is only elimination done once, in general letters.

Worked Example 1: Rice and Dal Prices (Cramer’s Rule)

A family in Delhi buys 3 kg of rice and 2 kg of dal for Rs. 360. Another week they buy 5 kg of rice and 1 kg of dal for Rs. 320. Let rice cost x per kg and dal cost y per kg. (The prices are for practice only.)

The equations are 3x + 2y = 360 and 5x + y = 320. Here a = 3, b = 2, c = 360, d = 5, e = 1, f = 320.

1. D = ae – bd = (3)(1) – (2)(5) = 3 – 10 = -7.
2. Dx = ce – bf = (360)(1) – (2)(320) = 360 – 640 = -280.
3. Dy = af – cd = (3)(320) – (360)(5) = 960 – 1,800 = -840.
4. x = -280 / -7 = 40.
5. y = -840 / -7 = 120.

Rice costs Rs. 40 per kg and dal costs Rs. 120 per kg. Check: 3(40) + 2(120) = 120 + 240 = 360, and 5(40) + 120 = 320. Both are correct.

Key takeaway: If D is zero, you cannot divide, and the system has either no solution or infinitely many. Always compute D first.

The Ratio Test from NCERT

Without solving, you can predict the nature of the solution by comparing ratios of coefficients. For a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0:

Ratio condition	Lines	Solutions
a1/a2 is not equal to b1/b2	Intersect	Exactly one
a1/a2 = b1/b2, not equal to c1/c2	Parallel	None
a1/a2 = b1/b2 = c1/c2	Same line	Infinitely many

Take 2x + 3y = 9 and 4x + 6y = 18. Here 2/4 = 3/6 = 9/18, so the lines coincide and there are infinite solutions. Change the second to 4x + 6y = 20, and the constants no longer match, so the lines are parallel and there is no solution. For 2x + 3y = 9 and 3x – y = 5, we compare 2/3 and 3/(-1), which differ, so there is one solution.

Worked Example 2: Boat on the Ganga (Elimination)

A boat travels 30 km downstream in 3 hours and the same 30 km upstream in 5 hours. Find the speed of the boat in still water and the speed of the current.

Let the boat speed be x km/h and the current y km/h. Downstream speed is x + y, and upstream speed is x – y.

1. Downstream speed = 30 / 3 = 10, so x + y = 10.
2. Upstream speed = 30 / 5 = 6, so x – y = 6.
3. Add the two equations: 2x = 16, so x = 8.
4. Put x = 8 in the first: 8 + y = 10, so y = 2.

The boat moves at 8 km/h in still water, and the current flows at 2 km/h. Questions like this are common in CBSE Class 10 and in quantitative sections of competitive exams.

Worked Example 3: Two Items with Different GST Rates

A shop in Pune sells two items on one bill. The total price before tax is Rs. 10,000. The first item carries 5 percent GST and the second carries 18 percent GST. The total GST on the bill is Rs. 1,020. Find the pre-tax price of each item.

1. Let x be the first item’s price and y the second’s. Then x + y = 10,000.
2. GST gives 0.05x + 0.18y = 1,020.
3. Substitute x = 10,000 – y: 0.05(10,000 – y) + 0.18y = 1,020.
4. Expand: 500 – 0.05y + 0.18y = 1,020, so 0.13y = 520.
5. Solve: y = 520 / 0.13 = 4,000. Then x = 6,000.

The 5 percent item costs Rs. 6,000 and the 18 percent item costs Rs. 4,000 before tax. Check: 5 percent of 6,000 is 300, and 18 percent of 4,000 is 720. The sum is 1,020. Official GST rates are listed on the GST portal, and they can change, so use the current slab when you work on a real bill.

Matrix Form of the Formula

The pair can also be written as a matrix equation. The coefficients go in a square block, the unknowns in a column, and the constants in another column. Solving means finding the unknown column.

For two variables, the matrix method gives the same result as Cramer’s rule. Its real strength shows when there are three, four or more unknowns, where a calculator or computer does the heavy work. A simultaneous equations solver uses exactly these formulas in the background.

Extending the Idea to Three Unknowns

With three unknowns x, y and z, you need three equations. The plan stays the same. Use two equations to remove z, use another two to remove z again, and you are left with a pair in x and y. Solve the pair, then find z.

Cramer’s rule also extends, with 3 by 3 determinants. Most students meet this in Class 12 and in JEE preparation. The steps are longer, so tools help a lot to avoid arithmetic slips.

Reading the Formula in Plain Words

Formulas look scary until you say them aloud. Cramer’s rule says this: to find x, take the constants and swap them in place of the x coefficients. Compute the cross-difference. Then divide by the cross-difference of the original coefficients.

The bottom number D is the same for both unknowns. It measures how different the two equations are from each other. If the two lines point in nearly the same direction, D is tiny, and the answer becomes very sensitive to small changes. If they point in the same direction exactly, D is zero, and no single crossing point exists.

This idea matters in real life. When two plans, two rate cards or two sets of readings are almost the same, small measurement errors can produce wildly different answers. That is one reason engineers worry about the size of D in their calculations.

Choosing Between the Formulas

• Use substitution when a variable has coefficient 1 or minus 1, as in the rice and dal pair.
• Use elimination when coefficients line up easily, as in the boat problem where adding cancels y.
• Use Cramer’s rule when the coefficients are awkward numbers, or when you plan to repeat the steps many times.
• Use the ratio test first if the question only asks whether a solution exists.

A good habit is to try the ratio test on every new pair. It takes a few seconds, and it protects you from wasting ten minutes on a system that has no solution.

Benefits of Knowing the Formulas

• Speed. A direct formula is quicker than long elimination for hard numbers.
• Prediction. The ratio test tells you what to expect before you start.
• Accuracy. Formulas are easy to program or to put in a spreadsheet.
• Understanding. You see that all methods are linked.
• Exam readiness. Board and entrance papers often reward the shortest correct route.

Challenges and Limitations

• Cramer’s rule fails when D is zero, so you need other reasoning in that case.
• Large determinants with decimals invite arithmetic mistakes.
• The formulas here are for linear equations only.
• Word problems still need careful translation into equations.
• Memorising without understanding makes it easy to mix up Dx and Dy.

Common Mistakes to Avoid

1. Swapping Dx and Dy. Dx replaces the x column with constants, and Dy replaces the y column.
2. Using the NCERT sign by mistake. In a1 x + b1 y + c1 = 0, the constant is on the left, so c is minus c1.
3. Forgetting to check D. Dividing by zero means the method does not apply.
4. Losing negative signs. Put brackets around negative numbers when substituting.
5. Mixing percentage and decimal. Write 18 percent as 0.18 in equations.
6. Skipping verification. One quick check shows if you slipped.

Best Practices

1. Name a, b, c, d, e and f clearly before applying the formulas.
2. Calculate D first, and stop if it is zero.
3. Use brackets for every negative number.
4. Keep decimals as fractions where it avoids rounding.
5. Verify the answer in both equations.
6. Confirm with a solver when the numbers are heavy.

Equations also help with property sums. If you know the total area of two rooms and how much bigger one is than the other, you can find each size with a pair of equations. A square footage calculator can then turn the result into sq ft, sq m or gaj for a quote.

Conclusion

Simultaneous equations rest on a few clean formulas. Elimination and substitution show you how the answer arises, and Cramer’s rule gives it in a single stroke. The ratio test tells you early whether a unique answer exists.

Practise each formula on one everyday problem, such as prices, speeds or tax bills. Once you see them work, the formulas stop being something to memorise and become tools you trust.

Frequently Asked Questions

What is the formula for solving simultaneous equations?

There is no single formula for every method, but Cramer’s rule is the direct one. For a x + b y = c and d x + e y = f, x = (ce – bf) / (ae – bd) and y = (af – cd) / (ae – bd). It only works when ae – bd is not zero.

What does it mean when the determinant D is zero?

It means the two lines are parallel or the same. Then the system has either no solution or infinitely many. Compare the constants to tell which case you have.

What is the NCERT condition for a unique solution?

In Class 10 NCERT, a pair a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 has a unique solution when a1/a2 is not equal to b1/b2. The lines then cross at exactly one point.

Is Cramer’s rule better than elimination?

Neither is always better. Cramer’s rule is a direct formula and suits repeated or awkward calculations. Elimination is often faster by hand when the numbers are small and simple.

Can these formulas handle decimals and fractions?

Yes. The same formulas work for any real numbers. Use brackets for negatives and keep fractions as fractions as long as possible to avoid rounding errors.