There are different ways to solve linear equations like 3x-4y=12, but the most common methods are substitution, elimination, graphing, and matrix inversion. In this guide, we will explain each method step by step and show how to obtain four different solutions for the equation.



The substitution method involves solving for one variable in terms of the other and substituting the expression into the other equation. Here's how to use the substitution method to solve 3x-4y=12:

Step 1: Solve for x or y in terms of the other variable. Let's solve for x in terms of y by adding 4y to both sides of the equation:

3x - 4y + 4y = 12 + 4y
3x = 12 + 4y
x = (12 + 4y)/3

Step 2: Substitute the expression for x into the other equation and simplify:

3x - 4y = 12
3((12 + 4y)/3) - 4y = 12
12 + 4y - 4y = 12
12 = 12

Step 3: Check if the solution is consistent or inconsistent. Since 12=12 is always true, the solution is consistent and has infinitely many solutions.



The elimination method involves adding or subtracting multiples of the equations to eliminate one variable. Here's how to use the elimination method to solve 3x-4y=12:

Step 1: Multiply one or both equations by constants to make the coefficients of one variable opposite in sign. Let's multiply the first equation by 4 and the second equation by 3 to eliminate y:

12x - 16y = 48
9x - 12y = 36

Step 2: Add or subtract the equations to eliminate the variable. Let's subtract the second equation from the first equation:

12x - 16y - (9x - 12y) = 48 - 36
3x - 4y = 12

Step 3: Solve for the remaining variable. We can use the substitution method or simply divide both sides by 3 to obtain:

x = (4y + 12)/3

Step 4: Check if the solution is consistent or inconsistent. Since the equation has one variable, it has infinitely many solutions.



The graphing method involves plotting the equations on a coordinate plane and finding the intersection point(s). Here's how to use the graphing method to solve 3x-4y=12:

Step 1: Rewrite the equation in slope-intercept form y=mx+b by solving for y:

3x - 4y = 12
-4y = -3x + 12
y = (3/4)x - 3

Step 2: Plot the line on a coordinate plane by finding two points or using the slope and y-intercept.

Step 3: Repeat

we can consider, that x=4 and y=0 then 3x-4y=12 3(4)-4(0)=12 12-0=12 12=12