- Linear equations is one of the foundation topics in the Quant section of CAT.
- Hence, the fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables.
- Be careful of silly mistakes in this topic as that is how students generally lose marks here.
- Generally, the number of equations needed to solve the given problem is equal to the number of variables.
What is a Linear Equation?
A linear equation is an equation which gives straight line when plotted on
- Linear equations can be of one variable or two variable or three variable.
- Let a, b, c and d are constants and x, y and z are variables. A general form of single variable linear equation is ax+b = 0.
- A general form of two variable linear equation is ax+by = c.
- A general form of three variable linear equation is ax+by+cz = d.
➢ Equations with Two Variables
- Consider two equations ax+by = c and mx+ny = p. Each of these equations represent two lines on the x-y co-ordinate plane. The solution of these equations is the point of intersection.
- If a / m = b / n ≠ c / p then the slope of the two equations is equal and so they are parallel to each other. Hence, no point of intersection occurs. Therefore no solution.
- If a / m ≠ b / n then the slope is different and so they intersect each other at a single point. Hence, it has a single solution.
- If a / m = b / n = c / p then the two lines are same and they have infinite points common to each other. So, infinite solutions occurs.
Try yourself:The pair of equations 3x – 5y = 7 and – 6x + 10y = 7 have:
a / m = b / n ≠ c / p
So these equations are parallel lines and hence they have no solution.
➢ General Procedure to Solve Linear Equations
- Aggregate the constant terms and variable terms.
- For equations with more than one variable, eliminate variables by substituting equations in their place.
- Hence, for two equations with two variables x and y, express y in terms of x and substitute this in the other equation.
- Example: Let x + y = 14 and x + 4y = 26 then x = 14-y (from equation 1) substituting this in equation 2, we get 14 - y + 4y = 26. Hence, y = 4 and x = 10.
- For equations of the form ax+by = c and mx+ny = p, find the LCM of b and n. Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.
- Example: Let 2x+3y = 13 and 3x+4y = 18 are the given equations (1) and (2).
⇒ LCM of 3 and 4 is 12.
⇒ Multiplying (1) by 4 and (2) by 3, we get 8x+12y = 52 and 9x+12y = 54.
⇒ (2) - (1) gives x = 2, y = 3
- If the system of equations has n variables with n-1 equations then the solution is indeterminate.
- If system of equations has n variables with n-1 equations with some additional conditions like the variables are integers then the solution may be determinate.
- If system of equations has n variables with n-1 equations then some combination of variables may be determinable.
- Example: If ax+by+cz = d and mx+ny+pz = q, if a, b, c are in Arithmetic progression and m, n and p are in AP then the sum x+y+z is determinable.
➢ Equations with Three Variables
- Let the equations be a1x + b1y + c1z = d1, a2x+b2y+c2z = d2 and a3x+b3y+c3z = d3 . Here we define the following matrices:
- If Determinant of D ≠ 0, then the equations have a unique solution.
- If Determinant of D = 0, and at least one but not all of the determinants Dx , Dy or Dz is zero, then no solution exists.
- If Determinant of D = 0, and all the three of the determinants Dx , Dy and Dz are zero, then there are infinitely many solution exists.
- Determinant can be calculated by D = a1(b2c3 -c2b3) - b1(a2c3 - c2a3) + c1(a2b3 -b2a3)