Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Certainly! Let's solve the given system of equations graphically step by step.
We have the following system of equations:
[tex]\[ \begin{array}{l} x + y = 0 \\ x - y + 2 = 0 \end{array} \][/tex]
### Step 1: Rearrange the Equations
First, let's rearrange the equations to express [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex] so we can graph them easily.
1. For the first equation [tex]\( x + y = 0 \)[/tex], we get:
[tex]\[ y = -x \][/tex]
2. For the second equation [tex]\( x - y + 2 = 0 \)[/tex], we get:
[tex]\[ x - y = -2 \implies -y = -x - 2 \implies y = x + 2 \][/tex]
So, our system now looks like:
[tex]\[ \begin{array}{l} y = -x \\ y = x + 2 \end{array} \][/tex]
### Step 2: Plot the Equations
Next, we can plot these two equations on graph paper.
Equation 1: [tex]\( y = -x \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. So, one point is (0, 0).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -1 \)[/tex]. So, another point is (1, -1).
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 1 \)[/tex]. Another point is (-1, 1).
Equation 2: [tex]\( y = x + 2 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. So, one point is (0, 2).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex]. Another point is (1, 3).
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 1 \)[/tex]. Another point is (-1, 1).
### Step 3: Draw the Lines
Using a straight edge, carefully draw the two lines based on the points:
- For [tex]\( y = -x \)[/tex]:
- Connect the points (0, 0), (1, -1), and (-1, 1).
- For [tex]\( y = x + 2 \)[/tex]:
- Connect the points (0, 2), (1, 3), and (-1, 1).
### Step 4: Find the Intersection
The solution to the system of equations is where the two lines intersect.
By looking at the points we noted:
- For [tex]\( y = -x \)[/tex]: the point (-1, 1)
- For [tex]\( y = x + 2 \)[/tex]: the point (-1, 1)
We can see that both lines intersect at the point [tex]\((-1, 1)\)[/tex].
### Solution Set
The solution to the system of equations is:
[tex]\[ (x, y) = (-1, 1) \][/tex]
So, your final solution set is [tex]\((-1, 1)\)[/tex].
Please make sure to neatly draw the graphs on graph paper, label each equation, and highlight the intersection point for clarity. Then submit your graph paper as instructed.
We have the following system of equations:
[tex]\[ \begin{array}{l} x + y = 0 \\ x - y + 2 = 0 \end{array} \][/tex]
### Step 1: Rearrange the Equations
First, let's rearrange the equations to express [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex] so we can graph them easily.
1. For the first equation [tex]\( x + y = 0 \)[/tex], we get:
[tex]\[ y = -x \][/tex]
2. For the second equation [tex]\( x - y + 2 = 0 \)[/tex], we get:
[tex]\[ x - y = -2 \implies -y = -x - 2 \implies y = x + 2 \][/tex]
So, our system now looks like:
[tex]\[ \begin{array}{l} y = -x \\ y = x + 2 \end{array} \][/tex]
### Step 2: Plot the Equations
Next, we can plot these two equations on graph paper.
Equation 1: [tex]\( y = -x \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. So, one point is (0, 0).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -1 \)[/tex]. So, another point is (1, -1).
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 1 \)[/tex]. Another point is (-1, 1).
Equation 2: [tex]\( y = x + 2 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. So, one point is (0, 2).
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex]. Another point is (1, 3).
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 1 \)[/tex]. Another point is (-1, 1).
### Step 3: Draw the Lines
Using a straight edge, carefully draw the two lines based on the points:
- For [tex]\( y = -x \)[/tex]:
- Connect the points (0, 0), (1, -1), and (-1, 1).
- For [tex]\( y = x + 2 \)[/tex]:
- Connect the points (0, 2), (1, 3), and (-1, 1).
### Step 4: Find the Intersection
The solution to the system of equations is where the two lines intersect.
By looking at the points we noted:
- For [tex]\( y = -x \)[/tex]: the point (-1, 1)
- For [tex]\( y = x + 2 \)[/tex]: the point (-1, 1)
We can see that both lines intersect at the point [tex]\((-1, 1)\)[/tex].
### Solution Set
The solution to the system of equations is:
[tex]\[ (x, y) = (-1, 1) \][/tex]
So, your final solution set is [tex]\((-1, 1)\)[/tex].
Please make sure to neatly draw the graphs on graph paper, label each equation, and highlight the intersection point for clarity. Then submit your graph paper as instructed.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
