Question

Review+the+graphs+of+a+system+of+these+two+linear+equations+in+two+variables Y=4x-10 and+y= -2+8+Find+the+solution+of+the+system

Answer 1

To analyze the system of equations given:

1. \( y = 4x - 10 \)
2. \( y = -2x + 8 \)

### Step 1: Graph the equations

**Equation 1: \( y = 4x - 10 \)**
- This is in slope-intercept form \( y = mx + b \) where \( m = 4 \) (slope) and \( b = -10 \) (y-intercept).
- The graph will cross the y-axis at \((0, -10)\).
- The slope \(4\) means for every 1 unit increase in \(x\), \(y\) increases by \(4\).

**Equation 2: \( y = -2x + 8 \)**
- Similarly, this is in slope-intercept form where \( m = -2 \) and \( b = 8 \).
- It will cross the y-axis at \((0, 8)\).
- The slope \(-2\) means for every 1 unit increase in \(x\), \(y\) decreases by \(2\).

### Step 2: Find the solution

To find the solution to the system, set the two equations equal to each other:

\[
4x - 10 = -2x + 8
\]

Now solve for \(x\):

1. Add \(2x\) to both sides:
\[
4x + 2x - 10 = 8
\]
\[
6x - 10 = 8
\]

2. Add \(10\) to both sides:
\[
6x = 18
\]

3. Divide by \(6\):
\[
x = 3
\]

Now, substitute \(x = 3\) back into one of the original equations to find \(y\). Using \(y = 4x - 10\):

\[
y = 4(3) - 10 = 12 - 10 = 2
\]

### Final Solution

The solution to the system of equations is:

\[
(x, y) = (3, 2)
\]

### Verification

To verify, substitute \(x = 3\) into the second equation \(y = -2x + 8\):

\[
y = -2(3) + 8 = -6 + 8 = 2
\]

Both equations yield \(y = 2\) at \(x = 3\).

Thus, the point of intersection is \((3, 2)\), which is the solution to the system of equations.