What is the first step for solving this system by substitution? DO NOT SOLVE!!

3x-2y=8
x+7y=10

How many solutions does this system of equations have?
3y-6x=12
y-2x=4
My answer is 1 solution. Is this correct?

No. Two equations in two unknows generally has one (x,y) solution. An exception occurs when the two equations are the same line, or parallel lines. Your example is one of the exceptions, since if you divide both sides of the first equation by 3, you get the second equation. There are an infinite number of x,y solutions

To solve a system of equations by substitution, the first step is to isolate one of the variables in one of the equations.

For the given system:
3x - 2y = 8
x + 7y = 10

To isolate a variable, you can select either equation and solve for that variable in terms of the other variable.

Let's take the second equation, x + 7y = 10, and solve for x:
x = 10 - 7y

Now that we have x in terms of y, we can substitute this expression into the first equation:
3(10 - 7y) - 2y = 8

This will create a new equation with only one variable, which can be solved to find the solution(s) to the system.

Regarding the question about the number of solutions for the second system:
3y - 6x = 12
y - 2x = 4

To determine the number of solutions, we need to analyze the equations. Two linear equations can have three possible outcomes:

1. They intersect at exactly one point, which means they have one unique solution.
2. They are parallel, meaning they have no common intersection point and no solution.
3. They are the same line, meaning they have infinitely many solutions.

To determine the number of solutions, we need to check the slopes of the two lines formed by the equations. If the slopes are equal and the y-intercepts are different, the lines are parallel and have no solution. If the slopes are equal and the y-intercepts are also equal, the equations represent the same line and have infinitely many solutions. If the slopes are not equal, the lines will intersect at exactly one point and have one unique solution.

In the given system, the two equations have different slopes (3 and -2) and different y-intercepts (12 and 4). Therefore, the system has one unique solution.

Your answer of 1 solution for the second system is correct.