Answer :
- The problem requires solving the equation $-\frac{1}{2}x + 4 = x + 1$ using a graph.
- Identify the intersection point of the lines $y = -\frac{1}{2}x + 4$ and $y = x + 1$ on the graph.
- Determine the x-coordinate of the intersection point, which is the solution to the equation.
- The solution is $x = 2$, which can be verified by substituting it back into the original equations. The final answer is $\boxed{2}$.
### Explanation
1. Understanding the Problem
The problem asks us to find the solution to the equation $-\frac{1}{2}x + 4 = x + 1$ using the graph of the two lines $y = -\frac{1}{2}x + 4$ and $y = x + 1$. The solution to this equation is the x-coordinate of the point where the two lines intersect.
2. Identifying the Intersection Point
Looking at the graph, we can see that the two lines intersect at the point (2, 3).
3. Determining the Solution
The x-coordinate of the intersection point (2, 3) is 2. Therefore, the solution to the equation $-\frac{1}{2}x + 4 = x + 1$ is $x = 2$.
4. Verification of the Solution
To verify the solution, we can substitute $x = 2$ into both equations:
For the first equation: $y = -\frac{1}{2}(2) + 4 = -1 + 4 = 3$
For the second equation: $y = (2) + 1 = 3$
Since both equations give $y = 3$ when $x = 2$, the solution is correct.
5. Final Answer
The solution to the equation $-\frac{1}{2} x+4=x+1$ is $x=2$.
### Examples
Imagine you and a friend are saving money. Your savings can be represented by the equation $y = -\frac{1}{2}x + 4$, where $y$ is your savings and $x$ is the number of weeks. Your friend's savings are represented by $y = x + 1$. The point where the two lines intersect (the solution to the equation) tells you the week when you and your friend have the same amount of savings. This concept is useful in comparing different financial plans or tracking progress towards a common goal.
- Identify the intersection point of the lines $y = -\frac{1}{2}x + 4$ and $y = x + 1$ on the graph.
- Determine the x-coordinate of the intersection point, which is the solution to the equation.
- The solution is $x = 2$, which can be verified by substituting it back into the original equations. The final answer is $\boxed{2}$.
### Explanation
1. Understanding the Problem
The problem asks us to find the solution to the equation $-\frac{1}{2}x + 4 = x + 1$ using the graph of the two lines $y = -\frac{1}{2}x + 4$ and $y = x + 1$. The solution to this equation is the x-coordinate of the point where the two lines intersect.
2. Identifying the Intersection Point
Looking at the graph, we can see that the two lines intersect at the point (2, 3).
3. Determining the Solution
The x-coordinate of the intersection point (2, 3) is 2. Therefore, the solution to the equation $-\frac{1}{2}x + 4 = x + 1$ is $x = 2$.
4. Verification of the Solution
To verify the solution, we can substitute $x = 2$ into both equations:
For the first equation: $y = -\frac{1}{2}(2) + 4 = -1 + 4 = 3$
For the second equation: $y = (2) + 1 = 3$
Since both equations give $y = 3$ when $x = 2$, the solution is correct.
5. Final Answer
The solution to the equation $-\frac{1}{2} x+4=x+1$ is $x=2$.
### Examples
Imagine you and a friend are saving money. Your savings can be represented by the equation $y = -\frac{1}{2}x + 4$, where $y$ is your savings and $x$ is the number of weeks. Your friend's savings are represented by $y = x + 1$. The point where the two lines intersect (the solution to the equation) tells you the week when you and your friend have the same amount of savings. This concept is useful in comparing different financial plans or tracking progress towards a common goal.
