Is it correct to say: Using Cramer's rule directly on equations written as ax + by + c = 0?

Move the constants to the right first: ax + by = −c. The constant column in Dx and Dy must be the right-hand side values.

Is it correct to say: Replacing the wrong column when forming Dx or Dy?

Dx: replace the x-column (first column) by the constants. Dy: replace the y-column (second column) by the constants.

Is it correct to say: Sign errors while expanding a determinant with negative entries?

Write each product in brackets, e.g. (2)(−2) − (3)(5). The rule is ad − bc, never ad + bc.

Is it correct to say: Stopping at m and n in equations reducible to linear form?

m and n are only helper variables. Finish with x = 1/m and y = 1/n (or solve x + y = 1/p, x − y = 1/q).

Is it correct to say: Applying Cramer's rule when D = 0?

Division by zero is not allowed. If D = 0 the equations have no unique solution — the lines are parallel or coincident.

Is it correct to say: Reading the intersection point carelessly from a rough graph?

Use a proper scale, plot at least four points per line, and verify the intersection point by substituting it in both equations.

📄 Formula sheet (PDF)

Linear Equations in Two Variables

Graphical method, elimination and substitution, determinants, Cramer's rule, equations reducible to linear form and word problems — Maharashtra SSC Std X Algebra Ch 1

📐 16 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

💡

Board Exam Tips

→Before using Cramer's rule, rewrite both equations in the form ax + by = c, with the constant on the right side. Otherwise Dx and Dy come out with the wrong sign.
→Write D, Dx and Dy as determinants first, then expand them. Each line earns method marks.
→For the graphical method, make a table of at least four ordered pairs for each line, as the textbook advises; the extra points catch plotting errors. Write the solution as an ordered pair (x, y) read from the point of intersection.
→In equations reducible to linear form, state the substitution (for example 1/x = m) and remember to convert back to x and y at the end.
→If the coefficients of x and y are interchanged in the two equations, add and subtract the equations to get x + y and x − y quickly.
→Always check your answer by substituting it in both original equations — it takes less than a minute.

📐 Formulas(16)

General Form of a Linear Equation in Two Variables

a x + b y + c = 0

Symbol	Meaning
$x, y$	Variables (unknowns)
$a, b$	Coefficients of x and y (not both zero)
$c$	Constant term

Simultaneous Equations in Standard Form

a_{1} x + b_{1} y = c_{1}, a_{2} x + b_{2} y = c_{2}

Points for Drawing the Graph

x = 0 \Rightarrow y = \frac{c}{b}, y = 0 \Rightarrow x = \frac{c}{a}

Graphical Solution★ Board fav

Solution = (x, y) = point of intersection of the two lines

Substitution Method

y = \frac{c _{1} - a _{1} x}{b _{1}}, b_{1} \neq = 0

Elimination Method (eliminating y)

(a_{1} b_{2} - a_{2} b_{1}) x = c_{1} b_{2} - c_{2} b_{1}

Interchanged Coefficients (Add and Subtract)

a x + b y = c, b x + a y = d \Rightarrow x + y = \frac{c + d}{a + b}, x - y = \frac{c - d}{a - b}

Determinant of Order 2★ Board fav

a c b d = a d - b c

Determinant D

D = a_{1} a_{2} b_{1} b_{2} = a_{1} b_{2} - a_{2} b_{1}

Determinant Dx

D_{x} = c_{1} c_{2} b_{1} b_{2} = c_{1} b_{2} - c_{2} b_{1}

Determinant Dy

D_{y} = a_{1} a_{2} c_{1} c_{2} = a_{1} c_{2} - a_{2} c_{1}

Cramer's Rule★ Board fav

x = \frac{D _{x}}{D}, y = \frac{D _{y}}{D}, D \neq = 0

Symbol	Meaning
$D$	Determinant of the coefficients of x and y
$D_{x}$	D with the x-column replaced by the constants
$D_{y}$	D with the y-column replaced by the constants

When Cramer's Rule Cannot Be Used

D = 0 \Rightarrow no unique solution

Reducible Form: Reciprocals of x and y★ Board fav

\frac{a _{1}}{x} + \frac{b _{1}}{y} = c_{1} \Rightarrow a_{1} m + b_{1} n = c_{1}, m = \frac{1}{x}, n = \frac{1}{y}

Reducible Form: x + y and x − y in Denominators

p = \frac{1}{x + y}, q = \frac{1}{x - y} \Rightarrow x + y = \frac{1}{p}, x - y = \frac{1}{q}

Two-Digit Number

N = 10 x + y, N_{reversed} = 10 y + x

Symbol	Meaning
$x$	Digit in tens place
$y$	Digit in units place

✏️ Solved Examples

1Solved Exampleeasy2 steps

Find the value of the determinant with first row (5, 3) and second row (−7, −4).

Write the determinant and use ad − bc.

5 - 7 3 - 4 = 5 \times (- 4) - 3 \times (- 7)

2Solved Exampleboard6 steps

Solve using Cramer's rule: 2x + 3y = 13 and 5x − 2y = 4.

Compare with a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

a_{1} = 2, b_{1} = 3, c_{1} = 13; a_{2} = 5, b_{2} = - 2, c_{2} = 4

3Solved Exampleboard5 steps

Solve: 2/x + 3/y = 13 and 5/x − 4/y = −2.

Put 1/x = m and 1/y = n.

2 m + 3 n = 13 (1), 5 m - 4 n = - 2 (2)

4Solved ExampleHOTS6 steps

Solve: 10/(x + y) + 6/(x − y) = 4 and 20/(x + y) − 3/(x − y) = 3.

Put 1/(x + y) = p and 1/(x − y) = q.

10 p + 6 q = 4 (1), 20 p - 3 q = 3 (2)

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using Cramer's rule directly on equations written as ax + by + c = 0
✓Move the constants to the right first: ax + by = −c. The constant column in Dx and Dy must be the right-hand side values.
2
Replacing the wrong column when forming Dx or Dy
✓Dx: replace the x-column (first column) by the constants. Dy: replace the y-column (second column) by the constants.
3
Sign errors while expanding a determinant with negative entries
✓Write each product in brackets, e.g. (2)(−2) − (3)(5). The rule is ad − bc, never ad + bc.
4
Stopping at m and n in equations reducible to linear form
✓m and n are only helper variables. Finish with x = 1/m and y = 1/n (or solve x + y = 1/p, x − y = 1/q).
5
Applying Cramer's rule when D = 0
✓Division by zero is not allowed. If D = 0 the equations have no unique solution — the lines are parallel or coincident.
6
Reading the intersection point carelessly from a rough graph
✓Use a proper scale, plot at least four points per line, and verify the intersection point by substituting it in both equations.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the value of the determinant with first row (4, −2) and second row (3, 5).
Q2
Solve using Cramer's rule: 3x + 2y = 10 and x − 3y = 7.
Q3
Solve: 37x + 43y = 123 and 43x + 37y = 117.
Q4
Solve: 4/x + 6/y = 7 and 3/x − 2/y = 2.
Q5
A two-digit number is 4 times the sum of its digits. The number obtained by reversing its digits is 18 more than the original number. Find the number.
Q6
Can the equations 2x + 3y = 5 and 4x + 6y = 7 be solved by Cramer's rule? Give a reason.

📝 Notes

Linear Equations in Two Variables

This chapter is about solving a pair of linear equations — finding the one ordered pair (x, y) that satisfies both. The Std X textbook adds two tools to the elimination and substitution methods you already know: the graphical method and Cramer's rule using determinants.

Choosing a method

Graphical method: asked when the question says "solve graphically". Draw both lines on one graph; the point of intersection is the solution.
Elimination / substitution: quickest for most pairs of equations. Make the coefficients of one variable equal, then add or subtract.
Cramer's rule: use when the question asks for it, or when the coefficients are awkward. It is mechanical and easy to check.

Cramer's rule step by step

Write the equations as $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ .
Find $D$ from the coefficients of x and y.
Find $D_x$ (constants in the x-column) and $D_y$ (constants in the y-column).
Write $x = D_x/D$ and $y = D_y/D$ , then check in both equations.

If $D = 0$ , stop — the equations do not have a unique solution.

Equations reducible to linear form

When x and y appear in denominators, such as $\frac{2}{x} + \frac{3}{y} = 13$ , substitute $m = \frac{1}{x}$ and $n = \frac{1}{y}$ . The new equations are linear, so any method works. The same idea works for $\frac{1}{x+y}$ and $\frac{1}{x-y}$ . The step students most often skip is converting back to x and y.

Word problems

Choose two unknowns, write two equations from the two conditions, and solve. Common settings are ages, two-digit numbers ( $10x + y$ ), fractions, perimeters of rectangles and money. Always state what x and y stand for and give the answer in words.