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Without the graphing, how do you decide whether the following system of linear equations has one solution, infinitely many solutions or no solution?

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Last updated date: 27th Feb 2024
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IVSAT 2024
Answer
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Hint:In order to explain the answer to this question, first we need to change the pair of linear equations in the slope-intercept form of equation. Then after written in slope-intercept form, we have to compare the slope of both the linear equations given. If the slope of both the equations are the same then they will either be parallel lines with no solution or the same line with infinite solution.

Complete step by step solution:
To solve the system of linear equations without graphing, you can use the substitution method. To solve the system of linear equations, first we solve one of the linear equations for one of the variables in the terms of other variables. Then the value we got, substituting this value for the same variable in the other equation and now solving for the other variable.
You can choose any of the linear equations first, it does not affect your answer.
For example:
Let, we have given a pair of linear equations.
\[\Rightarrow 4x-6y=-4\] and----- (1)
\[\Rightarrow 8x+2y=48\] ------ (2)
First step is to choose any of the equations and solve for any of the variables ‘x’ or ‘y’.
Taking equation (1) and solve for the value of ‘y’, we get
\[\Rightarrow 4x-6y=-4\]
\[\Rightarrow 4x=-4+6y\]
\[\Rightarrow x=\dfrac{-4+6y}{4}\]
Substitute the value of ‘x’ in the equation (2)
\[\Rightarrow 8\left( \dfrac{-4+6y}{4} \right)+2y=48\]
\[\Rightarrow 2\left( -4+6y \right)+2y=48\]
Solving the equation for ‘y’, we get
\[\Rightarrow -8+12y+2y=48\]
\[\Rightarrow -8+14y=48\]
\[\Rightarrow 14y=48+8\]
\[\Rightarrow 14y=56\]
\[\Rightarrow y=4\]
And substitute y=4 in\[x=\dfrac{-4+6y}{4}\], we get
\[\Rightarrow x=\dfrac{-4+6\times 4}{4}=\dfrac{-4+24}{4}=\dfrac{20}{4}=5\]
\[\Rightarrow x=5\]
Therefore, the solution of this pair of linear equations is x=5, y=4. This can also be written as (x, y) = (5, 4).
\[\Rightarrow \]To know whether the system of linear equation has many solution or no solution:
First we have to change both linear equation in the slope-intercept form i.e. y = mx + b
Compare the slope i.e. ‘m’ of both the equations.
\[\Rightarrow \]If the slope of both the linear equations are the same but the y-intercept are different, then the system of linear equations has no solution.
\[\Rightarrow \]If the slope of both the linear equations are different, then the system of linear equations has one solution.
\[\Rightarrow \]If the slope of both the linear equations given are same and the y-intercept of both the equations are also same, then the system of linear equations has infinite solution.

Note:
While explaining the answer to this question, students should remember these points:
If we get exact solution for ‘x’ and ‘y’ after solving system, then the system of linear equation
has one solution. If we get equality of two numbers i.e. for example; 3=3, then the system of linear equations has infinite solutions.
If we get inequality of two numbers i.e. for example; 5=3, then the system has no solution.