medium difficulty6 min

Algebraic solution by cross-multiplication

Cross-multiplication is a method used to solve a pair of linear equations written in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.

In one line: Standard form first, then cross carefully.

Practice This Concept

Understand it

This method uses ratios formed from the coefficients and constants to directly write the values of x and y. It is especially useful when the pair is already in standard form. Students should understand the formula and apply the signs carefully because one wrong sign changes the answer.

Example

For 2x + 3y - 7 = 0 and 3x - 2y - 4 = 0, the cross-multiplication formula can be used to find x and y.

Formula and rule

If a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then x = (b1c2 - b2c1)/(a1b2 - a2b1) and y = (c1a2 - c2a1)/(a1b2 - a2b1).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1For 2x + 3y - 7 = 0 and 3x - 2y - 4 = 0, the denominator is 2(-2) - 3(3) = -13.
2Then x = [3(-4) - (-2)(-7)] / -13 = 2, and y = [(-7)(3) - (-4)(2)] / -13 = 1.

Method you can reuse

1Put both equations in ax + by + c = 0 form.
2Write a1, b1, c1 and a2, b2, c2 carefully.
3Apply the x and y formulas with signs.
4Simplify the fractions.
5Verify the final values in both equations.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is cross-multiplication used only after writing equations in standard form?

Think first, then show answer

It is used after standard form because the formula depends on the coefficients a, b, and c being identified clearly. If the equations are not arranged properly, the sign and position errors become common.

Common wrong answer

A common wrong answer is to swap b1 and b2 or ignore the minus signs in c1 and c2. That gives a numerically neat but wrong answer.

Don't mix this up

Students often place the coefficients in the wrong positions in the formula or miss the minus signs in the numerators.

Exam tip

Write the coefficients neatly first and keep the standard form consistent before applying the formula.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

The exam may ask you to solve the pair by cross-multiplication or to identify the standard form needed for this method.

1
Enough for 1 mark
Cross-multiplication is a method used to solve a pair of linear equations written in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
2
Add for 2 marks
Example: For 2x + 3y - 7 = 0 and 3x - 2y - 4 = 0, the cross-multiplication formula can be used to find x and y.
3
Add for 3 marks
This method uses ratios formed from the coefficients and constants to directly write the values of x and y.

MCQ Quiz

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