Linear equation with two variables

An equation that has two variables: $X$ and $Y$ .
$y=a\times x+b$
To solve a linear equation that has two variables, we must find a pair of values for $X$ and for $Y$ that preserve the equation.
How will we do it?

Try to isolate one variable, whichever you prefer, then leave it alone on one side so that it does not have a value by itself.
Place any number you want instead of the variable you have not isolated and discover the value of the isolated variable.

In this way, you will be able to discover the pair of variables that satisfy the equation in question.

This type of equations generally has infinite solutions.
If you create a value table for this equation and treat it as a function, you can plot it on the Cartesian plane and see what it looks like graphically.

Mathematical representation of a linear equation in two variables: y = ax + b. A foundational concept in algebra, demonstrating the slope-intercept form, where 'a' represents the slope and 'b' the y-intercept. Featured in a guide on solving linear equations with two variables.

Detailed explanation

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Test yourself on linear equations with two variables!

$$ x+y=8 $$

$$ x-y=6 $$

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Example of a solution for a linear equation with two variables:
$x+6y=12$
Let's isolate the $X$
$x=12-6y$
Let's substitute any number for $Y$ , for example and find out the $X$ :
$y=4$
$x=12-6\times4$

$x=-12$
Notice that the result obtained is correct $x=-12, y=4$ ,
but, this is one among the infinite possible solutions for this equation.

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Examples and exercises with solutions of linear equation with two variables

Exercise #1

Solve the following system of equations:

$\begin{cases} x+y=8 \\ x=5-y \end{cases}$

Video Solution

Step-by-Step Solution

Note that in the current system of equations, one of the variables is isolated alone on the left side of the equation:

$\begin{cases} \underline{x}+y=8 \\ \bm{x=\underline{5-y}} \\ \end{cases}$

Therefore, we can apply the substitution method and substitute the entire expression that x equals in the second equation in place of x in the first equation (marked with an underline in both equations above) Hence we obtain one equation with one variable:

$\underline{ 5-y}+y=8$

Highlight the equation in which the variable we substituted is isolated in order to return to it later.

From here - we'll proceed to solve the single-variable equation that we obtained.

First- combine like terms on the left side of the resulting equation:

$5-y+y=8 \\ 5=8$

Note that y cancelled out in the current equation and we obtained a false statement, as shown below:

$5\neq8$ meaning-

We obtained a false statement regardless of the variables' values,

We can conclude from here that the system of equations has no solution, given that no matter which values we substitute for the variables - we won't obtain a true statement in both equations together.

Therefore the correct answer is answer D.