Question

Use the substitution method to solve the system y=−2x+1
𝑦
=
−
2
𝑥
+
1
, −4x=−2y+10
−
4
𝑥
=
−
2
𝑦
+
10

Answer 1

To solve the given system of equations using the substitution method, we first have the two equations:

1. \( y = -2x + 1 \) (Equation 1)
2. \( -4x = -2y + 10 \) (Equation 2)

We can start by substituting the expression for \( y \) from Equation 1 into Equation 2.

### Step 1: Substitute Equation 1 into Equation 2

Substituting \( y = -2x + 1 \) into Equation 2:

\[
-4x = -2(-2x + 1) + 10
\]

### Step 2: Simplify the equation

Now, we simplify it:

\[
-4x = -2(-2x) + (-2)(1) + 10 \\
-4x = 4x - 2 + 10
\]

This simplifies to:

\[
-4x = 4x + 8
\]

### Step 3: Solve for \( x \)

Now, let's add \( 4x \) to both sides:

\[
-4x + 4x = 4x + 4x + 8 \\
0 = 8x + 8
\]

Now, solve for \( x \):

\[
-8 = 8x \\
x = -1
\]

### Step 4: Substitute back to find \( y \)

Now that we have \( x = -1 \), we can substitute it back into Equation 1 to find \( y \):

\[
y = -2(-1) + 1 \\
y = 2 + 1 \\
y = 3
\]

### Final Solution

The solution to the system of equations is:

\[
(x, y) = (-1, 3)
\]

### Verification

To verify, we can check these values in both original equations:

1. Substitute into \( y = -2x + 1 \):

\[
3 = -2(-1) + 1 \Rightarrow 3 = 2 + 1 \Rightarrow 3 = 3 \quad \text{(True)}
\]

2. Substitute into \( -4x = -2y + 10 \):

\[
-4(-1) = -2(3) + 10 \Rightarrow 4 = -6 + 10 \Rightarrow 4 = 4 \quad \text{(True)}
\]

Both equations are satisfied, confirming that the solution \( (x, y) = (-1, 3) \) is correct.