CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables Notes Set 08

Linear Equation in Two Variables

1. An equation \(ax+by+c=0\) with \(a\) and \(b\) not both zero is a linear equation in two variables. Its graph is a straight line and every point on the line is a solution.
2. To draw it, express one variable in terms of the other, find at least two (preferably three) solutions, plot them and join them. A third point serves as a check.
3. The line meets the x-axis where \(y=0\) and the y-axis where \(x=0\). The graph of \(x=k\) is parallel to the y-axis and the graph of \(y=k\) is parallel to the x-axis.
4. A point lies on the line if its coordinates satisfy the equation.

Pair of Linear Equations

A pair is written as \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\). A solution of the pair is a point common to both lines.

When asked for a value of k, write the correct condition, solve for k, and then check the remaining ratio so that you do not confuse the parallel case with the coincident case.

Algebraic Methods

1. Substitution: express one variable from the simpler equation, substitute into the other, solve, and back-substitute.
2. Elimination: multiply the equations so that the coefficients of one variable are equal in magnitude, then add or subtract. When the coefficients are swapped (as in \(59x+47y\) and \(47x+59y\)), adding and subtracting directly is quickest.
3. Cross-multiplication: for \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), \(\dfrac{x}{b_1c_2-b_2c_1}=\dfrac{y}{c_1a_2-c_2a_1}=\dfrac{1}{a_1b_2-a_2b_1}\), valid when \(a_1b_2-a_2b_1\ne0\).
4. Equations in \(\dfrac1x\) and \(\dfrac1y\): put \(p=\dfrac1x\), \(q=\dfrac1y\), solve the linear pair, then invert. Equations with xy on the right can be divided by xy first.
5. Always substitute the answer into both original equations.

Word Problems

1. Choose the unknowns, translate each sentence into one equation, solve the pair and state the answer with units.
2. Digits: the number \(10a+b\) becomes \(10b+a\) when reversed; their sum is \(11(a+b)\) and their difference is \(9(a-b)\).
3. Ages: add or subtract the same number of years to both persons.
4. Fractions: let the fraction be \(\dfrac xy\) and apply each change to the numerator or denominator as stated.
5. Division: dividend = divisor x quotient + remainder.
6. Geometry: use the angle sum of 180 degrees for a triangle and 360 degrees for a quadrilateral; opposite angles of a cyclic quadrilateral add to 180 degrees.
7. Always verify that the answer satisfies the story and not only the equations.