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CBSE Class 10 · Mathematics

Pair of Linear Equations in Two Variables

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Pair of Linear Equations in Two Variables” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Graphical Method of Solution of a Pair of Linear Equations

A pair a1x + b1y = c1, a2x + b2y = c2 graphs as two straight lines. A solution is a point on both. Crossing once is one solution; drawing both and reading the intersection is the graphical method. A freehand miss of the crossing is not a solution.

Two curves that are not lines are not this pair.

Figure. Each equation is one line. Their crossing P is the only pair that satisfies both.

How it works

  1. Draw both lines from two points eachThe graph.
  2. Read the common point, if it is thereThe solution.
  3. If they miss or sit on top, name none or infinitely manyThe picture’s verdict.

2Algebraic Methods of Solving a Pair of Linear Equations

Algebraic methods find the same (x, y) without a scale-drawing: substitute, eliminate, or cross-multiply as taught. Each is a rewrite that keeps the pair honest. A method that changes only one equation’s meaning is a lie.

Guessing a pretty pair and skipping the rewrite is not a method.

Figure. Algebra finds the same meeting pair the graph marks. 5+3=8 and 5-3=2.

How it works

  1. Pick substitution, elimination, or cross-multiplyThe tool.
  2. Rewrite until one variable dropsThe algebra.
  3. Back-substitute and check in both originalsThe pair.

3Substitution method

Substitution: solve one equation for a variable, plug into the other. For x + y = 5 and x − y = 1, y = 5 − x into the second gives 2x = 6, x = 3, y = 2. The isolate must be exact; dropping a sign is a new pair.

Substituting into the same equation you solved is a loop, not a method.

Figure. Isolate one letter, replace it in the other sentence, then read the pair. 2(1)+3=5.

How it works

  1. Isolate one variable from one equationThe express.
  2. Plug into the otherThe drop.
  3. Find the second variable and check bothSubstitution.

x + y = 5, x − y = 1

Solve by substitution.

  • From firsty = 5 − x
  • Into secondx − (5 − x) = 1 → 2x = 6
  • (x, y)(3, 2)

Pro tip. Isolate, plug into the other, check both.

4Elimination method

Elimination: multiply so one variable’s coefficients match, then add or subtract to drop it. For the same pair, add: 2x = 6. The multiplier must hit every term. Subtracting the wrong way can double a variable instead of dropping it.

Cancelling x by striking it in only one equation is not elimination.

Figure. Same 2x in both rows. Subtract the rows and x vanishes. 2y=4 so y=2, then x=2.5.

How it works

  1. Match one variable’s coefficientsThe prepare.
  2. Add or subtract the equationsThe drop.
  3. Solve and back-substituteElimination.

5Cross-multiplication method

Cross-multiplication writes the school arrangement for two equations as a proportion that gives x and y (as taught). It is a compact form of the same pair, not a third kind of answer. If a denominator is 0, that case is the none-or-infinite heading, not a silent ÷0.

Cross-multiplying as if it were two fractions from a recipe, ignoring the pair’s layout, is a miss.

Figure. Cross-multiplication reads this 2-by-3 coefficient table. Each unknown is a crossed pair over the determinant.

How it works

  1. Write the two equations in standard formThe prepare.
  2. Apply the taught cross layoutThe method.
  3. If a taught denominator vanishes, sort none vs infinite instead of dividingHonest.

6A pair can have one solution, none, or infinitely many

A pair has one solution, none, or infinitely many. Compare a1/a2, b1/b2, c1/c2 as taught: different slope-ratios → one; same slope, different intercept-ratio → none; all three equal → infinite. The sort is a test, not a mood about “hard” equations.

Two messy-looking equations are not automatically “no solution”.

Figure. One crossing, two parallels, or the same line twice. Those are the three counts of solutions.

How it works

  1. Form the three ratiosThe compare.
  2. Name one / none / infiniteThe sort.
  3. Match the graph: cross / parallel / same lineThe picture agrees.

7Parallel lines never meet

Parallel lines never meet: same slope, different intercept. The algebraic none-case is this picture. A pair that looks parallel on a thick marker but has a crossing in the algebra is a drawing-miss, not a new law.

“They look like they will meet off the page” is not a solution.

Figure. Equal slopes, different intercepts. The lines stay a fixed height apart and never meet.

How it works

  1. See equal a1/a2 = b1/b2 but not equal to c1/c2The parallel test.
  2. Say no (x, y) sits on bothNone.
  3. Refuse an off-page fantasy pointNever meet.

8The same line is infinitely many solutions

The same line is infinitely many solutions: every point of the line satisfies both writings because they are the same condition twice. Infinite is a line of answers, not “any numbers you like” that miss the line.

Picking (0, 0) always as “infinite” is a miss if the line misses the origin.

Figure. The second equation is a multiple of the first, so both name one line. Every point on it is a solution.

How it works

  1. See a1/a2 = b1/b2 = c1/c2The same-line test.
  2. Name infinitely many — all points of that lineThe heading.
  3. One point on the line is a sample, not the whole setInfinite means the line.
x + y = 5 and x − y = 1 solve as
  1. (3, 2)
  2. (5, 1)
  3. No solution

Add: 2x = 6; y = 2.

Notes

  • The official chapter title is “Pair of Linear Equations in Two Variables”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • one / none / infinite from a1/a2, b1/b2, c1/c2 (as taught)

Recap

Hold these pegs from the official chapter “Pair of Linear Equations in Two Variables”. The wording is ExamMaster’s teaching, not a textbook recap.

Graphical Method of Solution of a Pair of Linear Equations
A pair a1x + b1y = c1, a2x + b2y = c2 graphs as two straight lines.
Algebraic Methods of Solving a Pair of Linear Equations
Algebraic methods find the same (x, y) without a scale-drawing: substitute, eliminate, or cross-multiply as taught.
Substitution method
Substitution: solve one equation for a variable, plug into the other.
Elimination method
Elimination: multiply so one variable’s coefficients match, then add or subtract to drop it.
Cross-multiplication method
Cross-multiplication writes the school arrangement for two equations as a proportion that gives x and y (as taught).
A pair can have one solution, none, or infinitely many
A pair has one solution, none, or infinitely many.

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