solve this using substitution, 4y-x=-3, 2x-8y=6



We can solve the system of equations using substitution. The two equations are:

1. \( 4y - x = -3 \)
2. \( 2x - 8y = 6 \)

### Step 1: Solve one equation for one variable
Let’s solve the first equation for \(x\) in terms of \(y\).

\[
4y - x = -3
\]
Solve for \(x\):
\[
x = 4y + 3
\]

### Step 2: Substitute this expression for \(x\) into the second equation
Now, substitute \(x = 4y + 3\) into the second equation \(2x - 8y = 6\):

\[
2(4y + 3) - 8y = 6
\]
Expand the equation:
\[
8y + 6 - 8y = 6
\]
Simplify:
\[
6 = 6
\]

This is a true statement, indicating that the system of equations is **dependent**, meaning the two equations represent the same line. Therefore, there are **infinitely many solutions**.

### Final Answer:
The system has infinitely many solutions, and the equations are dependent.

Would you like more details or have any questions?

Here are 5 related questions:
1. How can we identify when a system has no solutions?
2. What is the difference between dependent and independent systems?
3. Can substitution be used for non-linear systems?
4. How can graphing confirm that two lines are the same?
5. What is the elimination method, and when is it preferable?

**Tip:** Always check for special cases (like dependent or inconsistent systems) when solving systems of equations.

solve this using substitution, 4y-x=-3, 2x-8y=6

This detailed solution explains how to solve the system of linear equations 4y - x = -3 and 2x - 8y = 6 using the substitution method. It covers key algebraic concepts such as dependent systems and infinitely many solutions. Suitable for students in Grades 9-11.

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