Question

Write two solutions of \[3x + 4y = 7,x = 6y\] .

Answer 1

Hint:
Here, we have to find the solution set for the given equations. We will substitute the different values for one variable in the equation to find the other variables. We will follow the same method for both the given equations. Thus, we will obtain a solution set of two equations. An equation is defined as a mathematical statement with an equality sign between the two algebraic expressions.

Complete Step by step Solution:
We are given two equations \[3x + 4y = 7\] and \[x = 6y\].
Let us consider \[3x + 4y = 7\] ………………………………………………………………….\[\left( 1 \right)\]
Substituting \[x = 0\] in equation \[\left( 1 \right)\], we get
\[ \Rightarrow 3\left( 0 \right) + 4y = 7\]
By multiplying the terms, we get
\[ \Rightarrow 0 + 4y = 7\]
\[ \Rightarrow 4y = 7\]
Dividing both sides by 4, we get
\[ \Rightarrow y = \dfrac{7}{4}\]
Thus, the solution set is \[\left( {0,\dfrac{7}{4}} \right)\] by substituting \[x = 0\] in equation \[\left( 1 \right)\].
Substituting \[x = 1\] in equation \[\left( 1 \right)\], we get
\[3\left( 1 \right) + 4y = 7\]
By multiplying the terms, we get
\[ \Rightarrow 3 + 4y = 7\]
By rewriting the equation, we get
\[ \Rightarrow 4y = 7 - 3\]
\[ \Rightarrow 4y = 4\]
Dividing both sides by 4, we get
\[ \Rightarrow y = \dfrac{4}{4}\]
\[ \Rightarrow y = 1\]
Thus, the solution set is \[\left( {1,1} \right)\] by substituting \[x = 1\] in equation \[\left( 1 \right)\].
Now, consider an equation \[x = 6y\] ……………………………………………………………………………………………\[\left( 2 \right)\]
 Substituting \[x = 0\] in equation \[\left( 2 \right)\], we get
\[0 = 6y\]
Dividing both sides by 6, we get
\[ \Rightarrow y = 0\]
Thus, the solution set is \[\left( {0,0} \right)\] by substituting \[x = 0\] in equation \[\left( 2 \right)\].
Substituting \[x = 1\] in equation \[\left( 2 \right)\], we get
\[1 = 6y\]
By rewriting the equation, we get
\[ \Rightarrow 6y = 1\]
Dividing both sides by 6, we get
\[ \Rightarrow y = \dfrac{1}{6}\]
Thus, the solution set is \[\left( {1,\dfrac{1}{6}} \right)\] by substituting \[x = 1\] in equation \[\left( 2 \right)\].
Therefore, the two solutions set for the equation \[3x + 4y = 7\] are \[\left( {0,\dfrac{7}{4}} \right)\] and \[\left( {1,1} \right)\].

Also, the two solutions set for the equation \[x = 6y\] are \[\left( {0,0} \right)\]and \[\left( {1,\dfrac{1}{6}} \right)\].

Note:
We know that the solution set is a set of values which satisfies the relation between the two mathematical expressions. The solution set for the linear equations of two variables and with only one equation can be obtained only by the method of substitution. The variables can be substituted with either positive or negative integers. The solutions can be determined only by inspection.