Question

$y = 3x - 5\;{\text{and}}\;6x = 2y + 10$ How do I solve this?

Answer 1

First see if the lines are collinear lines or parallel lines or intersecting lines, two lines are said to be parallel if their ratios of coefficient of $x$ to coefficient of $y$ are equal, two lines are coincident if ratios of coefficient of $x$ to coefficient of $y$ are equal as well as the constant part too is equal and intersecting when ratios of coefficient of $x$ to coefficient of $y$ are not equal.
And parallel lines are inconsistent and have no solution, whereas collinear and intersecting lines have infinite and one solution respectively.

Complete step by step solution:
To solve the given equations $y = 3x - 5\;{\text{and}}\;6x = 2y + 10$,
we will first check whether the lines are parallel or collinear or intersecting,
If we see the equation of lines carefully, then we will find that the second equation is just multiplied with $2$ and has some balanced algebraic operation done which doesn’t affect the originality of the equation. We can see it as follows
$6x = 2y + 10 \\\ \Rightarrow 6x - 10 = 2y \\\ \Rightarrow 2 \times (3x - 5) = 2 \times y \\\ \Rightarrow 3x - 5 = y \\\ \Rightarrow y = 3x - 5 \\\$
So we have found that both the equations are similar to each other that mean both lines are collinear and have an infinite number of solutions, each point on the line is a solution for them. Just put any random real value of one variable and get the value for another, this will surely satisfy another equation.

Note: Solution of two equation of lines gives the coordinates of point where the lines met, and here in this question both lines are collinear, that’s why they have infinite number of solutions and we can find then by putting any random real value of one variable and get the value for another one from the equation.