CAT Solving linear Equations in two variables (equations of the form of ax + by +c = 0) MCQ

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Solve for x and y: $0.4 x-1.5 y=6.5,\;\; 0.3 x+0.2 y=0.9$

x = 5 and y = –3

x = –5 and y = –3

x = 5 and y = 3

None of the abve

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Concepts Covered - 1

Solving linear Equations in two variables (equations of the form of ax + by +c = 0)

Introduction:

A linear equation in two variables is represented by the equation:

$\mathrm{\[ ax + by + c = 0 \]}$

Where:

$\mathrm{- $ x $ and $ y $}$ are variables.

$\mathrm{- $ a $, $ b $, and $ c $ }$ are constants and $\mathrm{$ a $ and $ b $ }$ cannot both be zero.

The solution to this equation is an ordered pair (x,y) that satisfies the equation.

Method to Solve:

To solve such equations, you can use any of the following methods:

1. Substitution Method: Solve one of the equations for one variable in terms of the other variable. Then substitute this expression into the other equation.

2. Elimination Method: Multiply or divide one or both of the equations to eliminate one of the variables.

3. Graphical Method: Plot both lines on a graph. The point of intersection will be the solution.

Foundation Building Questions:

Question:

Given the equations:

1) $\mathrm{$ 2x + 3y = 6 $}$

2) $\mathrm{ $ x - 4y = 5 $}$

Find the values of $\mathrm{ $ x $ and $ y $.}$

Solution:

Using the Elimination Method:

Multiplying the second equation by 2, we get:

$\mathrm{$ 2x - 8y = 10 $}$ ... (i)

Subtracting (i) from the first equation, we get:

$\mathrm{$ 11y = -4 $}$

So, $\mathrm{$ y = -4/11 $}$

Substituting the value of $ y $ in the first equation:

$\mathrm{$ 2x + 3(-4/11) = 6 $}$

Solving, we get $\mathrm{$ x = 35/11 $.}$

So, the solution is $\mathrm{$ x = 35/11 $ and $ y = -4/11 $.}$

Tips and Tricks:

1. Quick Visualization: Before diving into calculations, do a quick mental check. If you can visualise the lines representing the equations, you might be able to predict if they intersect, are parallel, or coincide.

2. Zero Coefficients: If one of the coefficients is zero, it simplifies the equation significantly. For example, if $\mathrm{ $ b = 0 $}$ in $\mathrm{$ ax + by + c = 0 $}$ , the equation becomes a function of $\mathrm{$ x $ }$ only.

3. Consistent System: If two lines are represented by two different linear equations and they intersect at one point, it's a consistent system. If they're parallel, it's inconsistent.

4. Eliminate Wisely: In the elimination method, choose the variable which has coefficients closer in magnitude in both equations. This way, the elimination process becomes simpler.

5. Verify Solutions: Once you've found the values of $\mathrm{ $ x $ and $ y $}$ , quickly plug them back into the original equations to confirm they satisfy both.

Remember, consistent practice and solving various types of problems related to this concept will enhance your understanding and speed in solving them during the actual exam.