To analyze the system of equations and determine the number of solutions, we can start by rewriting the given equations, solving for one variable in terms of the other, and checking their relationships.
Step 1: Write the equations
- \( 2x = y + 4 \)
- \( -4x + 2y = 2 \)
Step 2: Rearranging the first equation
From the first equation, we can express \( y \) in terms of \( x \): \[ y = 2x - 4 \]
Step 3: Rearranging the second equation
Next, we can simplify the second equation: \[ -4x + 2y = 2 \] We can isolate \( y \) by first moving \( -4x \) to the right side: \[ 2y = 4x + 2 \] Now, divide by 2: \[ y = 2x + 1 \]
Step 4: Compare the two equations
We now have two equations in slope-intercept form:
- \( y = 2x - 4 \)
- \( y = 2x + 1 \)
Both lines have the same slope of \( 2 \). However, they have different \( y \)-intercepts (the first equation has \( -4 \) and the second has \( +1 \)). Because they have the same slope but different intercepts, these two lines are parallel.
Conclusion about the number of solutions
Since the lines are parallel, they will never intersect. Therefore, the system of equations has no solutions.
How to tell from the graph
If you were to graph these equations, you would plot both lines on the same coordinate plane.
- The first line would slope upwards with a slope of 2 and cross the y-axis at -4.
- The second line would also slope upwards with the same slope of 2, but would cross the y-axis at +1.
Since both lines are parallel, as stated earlier, they would never meet at any point, confirming that the system of equations has no solutions.
Summary
- From algebra: The lines have the same slope and different intercepts, indicating that they are parallel with no intersection points (no solutions).
- From the graph: Visual representation would show two parallel lines that do not intersect (no solutions).