The image shows two linear equations in two variables, represented graphically as straight lines on a coordinate plane. The equations are:
1. \( 3x + 2y = 6 \)
2. \( 3x + 2y = 12 \)
Problem:
The task is to solve the system of linear equations and interpret the solution geometrically.
---
Step-by-Step Solution:
####
Step 1: Analyze the Equations
Both equations are in the standard form of a linear equation: \( ax + by = c \).
- Equation 1: \( 3x + 2y = 6 \)
- Equation 2: \( 3x + 2y = 12 \)
####
Step 2: Compare the Slopes and Intercepts
To understand the relationship between these lines, we compare their slopes and intercepts.
1.
Slope-Intercept Form:
- Rewrite each equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For \( 3x + 2y = 6 \):
\[
2y = -3x + 6 \implies y = -\frac{3}{2}x + 3
\]
- Slope (\( m \)): \( -\frac{3}{2} \)
- y-intercept (\( b \)): \( 3 \)
For \( 3x + 2y = 12 \):
\[
2y = -3x + 12 \implies y = -\frac{3}{2}x + 6
\]
- Slope (\( m \)): \( -\frac{3}{2} \)
- y-intercept (\( b \)): \( 6 \)
2.
Comparison:
- Both lines have the same slope (\( -\frac{3}{2} \)), which means they are parallel.
- However, their y-intercepts are different (\( 3 \) and \( 6 \)), so they do not overlap.
####
Step 3: Geometric Interpretation
Since the lines are parallel and do not intersect, there is no point \((x, y)\) that satisfies both equations simultaneously. This means the system of equations has
no solution.
####
Step 4: Algebraic Verification
To confirm, let's try to solve the system algebraically by setting the equations equal to each other (since both are equal to \( 3x + 2y \)):
\[
3x + 2y = 6
\]
\[
3x + 2y = 12
\]
Subtract the first equation from the second:
\[
(3x + 2y) - (3x + 2y) = 12 - 6
\]
\[
0 = 6
\]
This is a contradiction, confirming that the system has no solution.
---
Final Answer:
The system of equations \( 3x + 2y = 6 \) and \( 3x + 2y = 12 \) has
no solution because the lines are parallel and do not intersect.
\[
\boxed{\text{No Solution}}
\]
Parent Tip: Review the logic above to help your child master the concept of linear equation in two variables worksheet.
