Question

How do you solve the system by graphing $2x+4y=2$ and $x+2y=1$ ?

Answer 1

Hint: $2x+4y=2$ and $x+2y=1$ are straight lines. A system of 2 straight lines $ax+by=c$ and ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ will have one solution if $\dfrac{a}{{{a}_{1}}}=\dfrac{b}{{{b}_{1}}}$ , the system will have no solution if $\dfrac{a}{{{a}_{1}}}=\dfrac{b}{{{b}_{1}}}\ne \dfrac{c}{{{c}_{1}}}$ ,
The system will have infinite solution if $\dfrac{a}{{{a}_{1}}}=\dfrac{b}{{{b}_{1}}}=\dfrac{c}{{{c}_{1}}}$. If $\dfrac{a}{{{a}_{1}}}=\dfrac{b}{{{b}_{1}}}=\dfrac{c}{{{c}_{1}}}$ then the two lines $ax+by=c$ and ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ are same one line will lie on other line.

Complete step by step answer:
The given 2 equations are $2x+4y=2$ and $x+2y=1$ , we know these 2 equations are straight line
If we compare $2x+4y=2$ to straight line $ax+by=c$ , then value of a is 2 , value of b is 4 and c is equal to 2
If we compare $x+2y=1$ to straight line ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ then the value of ${{a}_{1}}$ is equal to 1, the value of ${{b}_{1}}$ is 2 and the value of ${{c}_{1}}$ is 1
The value of $\dfrac{a}{{{a}_{1}}}=\dfrac{2}{1}=2$ , $\dfrac{b}{{{b}_{1}}}=\dfrac{4}{2}=2$ and $\dfrac{c}{{{c}_{1}}}=\dfrac{2}{1}=2$
We can see $\dfrac{a}{{{a}_{1}}}=\dfrac{b}{{{b}_{1}}}=\dfrac{c}{{{c}_{1}}}$ , so the given 2 straight lines are same
The point on the straight line $2x+4y=2$ will satisfy the equation $x+2y=1$ because both are same
We can see that from the graph

We can see one line overlap on another so there are infinite common points between the 2 lines.

Note:
Parallel lines never touch each other so there will be no common point between 2 parallel lines and slope of parallel lines are the same. Always remember the fact that if the slope of 2 lines in a 2 dimensional Cartesian plane are different then there always exists a common point between the 2 lines , they always intersect.