solve the system of linear equations: -2x+y=8,-4x+16=2y

Let's solve the system of linear equations:

\[
\begin{align*}
\text{Equation 1:} & \quad -2x + y = 8 \quad \text{(1)} \\
\text{Equation 2:} & \quad -4x + 16 = 2y \quad \text{(2)}
\end{align*}
\]

### Step 1: Simplify the Second Equation
First, let's simplify the second equation by dividing all terms by 2:

\[
-2x + 8 = y
\]

We can rewrite this as:

\[
y = -2x + 8 \quad \text{(3)}
\]

### Step 2: Substitution
Notice that equation (3) is the same as equation (1). Therefore, both equations represent the same line, meaning the system of equations has infinitely many solutions.

### Conclusion
Since both equations represent the same line, the system has infinitely many solutions. Any pair \((x, y)\) that satisfies one equation will satisfy the other.

### Solution
The solution can be expressed as:

\[
\boxed{y = -2x + 8}
\]

So, for any value of \(x\), \(y\) can be determined using the equation \(y = -2x + 8\).

Would you like more details, or do you have any questions?

---

Here are 5 related questions for further practice:
1. Solve the system: \(3x + y = 7\) and \(6x + 2y = 14\).
2. Determine whether the system \(x - 2y = 5\) and \(2x - 4y = 10\) has one solution, no solution, or infinitely many solutions.
3. Find the intersection point of the lines \(y = 3x + 1\) and \(y = -2x + 4\).
4. Solve for \(x\) and \(y\): \(x + y = 10\) and \(2x - y = 5\).
5. If \(x + y = 2\) and \(2x + 2y = 4\), determine the number of solutions.

**Tip:** When solving systems of linear equations, always check if the equations are multiples of each other. This can indicate if the system has infinitely many solutions or no solution at all.

Learn how to solve the system of linear equations -2x+y=8 and -4x+16=2y using the substitution method. This problem explores key algebraic concepts, including linear equations and their solutions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation