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To graph this system of linear equations and find the intersection point, we need to break down the process step-by-step. Here's how you can solve and graph the system:
### Step 1: Identify the equations
We have the following system of linear equations:
[tex]\[ \left\{ \begin{array}{l} y = x + 4 \\ 2x + y = 10 \end{array} \right. \][/tex]
### Step 2: Express both equations in slope-intercept form (if not already)
The first equation is already in slope-intercept form (y = mx + b):
[tex]\[ y = x + 4 \][/tex]
For the second equation:
[tex]\[ 2x + y = 10 \][/tex]
We can solve for [tex]\(y\)[/tex] to put it in slope-intercept form:
[tex]\[ y = 10 - 2x \][/tex]
### Step 3: Find the intersection point algebraically
To find the intersection point, we set the right sides of the equations equal to each other since at the point of intersection both [tex]\(y\)[/tex]-values will be the same:
[tex]\[ x + 4 = 10 - 2x \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 2x = 10 - 4 \\ 3x = 6 \\ x = 2 \][/tex]
Now substitute [tex]\(x = 2\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. Using [tex]\(y = x + 4\)[/tex]:
[tex]\[ y = 2 + 4 \\ y = 6 \][/tex]
So, the intersection point is [tex]\((2, 6)\)[/tex].
### Step 4: Plot the equations and intersection on a graph
1. Plot the first equation [tex]\(y = x + 4\)[/tex]:
- This line has a slope of 1 and y-intercept at [tex]\(y = 4\)[/tex].
- So, it will pass through points like [tex]\((0, 4)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((2, 6)\)[/tex], and so on.
2. Plot the second equation [tex]\(y = 10 - 2x\)[/tex]:
- This line has a slope of -2 and y-intercept at [tex]\(y = 10\)[/tex].
- So, it will pass through points like [tex]\((0, 10)\)[/tex], [tex]\((1, 8)\)[/tex], [tex]\((2, 6)\)[/tex], and so on.
When both lines are plotted, you will see they intersect at the point [tex]\((2, 6)\)[/tex].
### Step 5: Drawing the graph
To visualize:
1. Draw the coordinate axes.
2. Plot the line [tex]\(y = x + 4\)[/tex]. Start at the point [tex]\((0, 4)\)[/tex] and use the slope to find another point like [tex]\((1, 5)\)[/tex]. Draw the line through these points.
3. Plot the line [tex]\(y = 10 - 2x\)[/tex]. Start at the point [tex]\((0, 10)\)[/tex] and use the slope to find another point like [tex]\((1, 8)\)[/tex]. Draw the line through these points.
4. Mark the intersection point at [tex]\((2, 6)\)[/tex].
Your graph should look like this:
- A blue line (or some color) passing through [tex]\((0, 4)\)[/tex] with a slope of 1.
- A red line (or some color) passing through [tex]\( (0, 10) \)[/tex] with a slope of -2.
- They intersect at the point [tex]\((2, 6)\)[/tex].
### Step 6: Conclusion
From the graph and algebraic solution, the intersection point of the given system of linear equations is [tex]\((2, 6)\)[/tex].
This step-by-step method provides a thorough understanding of solving and graphing systems of linear equations.
### Step 1: Identify the equations
We have the following system of linear equations:
[tex]\[ \left\{ \begin{array}{l} y = x + 4 \\ 2x + y = 10 \end{array} \right. \][/tex]
### Step 2: Express both equations in slope-intercept form (if not already)
The first equation is already in slope-intercept form (y = mx + b):
[tex]\[ y = x + 4 \][/tex]
For the second equation:
[tex]\[ 2x + y = 10 \][/tex]
We can solve for [tex]\(y\)[/tex] to put it in slope-intercept form:
[tex]\[ y = 10 - 2x \][/tex]
### Step 3: Find the intersection point algebraically
To find the intersection point, we set the right sides of the equations equal to each other since at the point of intersection both [tex]\(y\)[/tex]-values will be the same:
[tex]\[ x + 4 = 10 - 2x \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 2x = 10 - 4 \\ 3x = 6 \\ x = 2 \][/tex]
Now substitute [tex]\(x = 2\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. Using [tex]\(y = x + 4\)[/tex]:
[tex]\[ y = 2 + 4 \\ y = 6 \][/tex]
So, the intersection point is [tex]\((2, 6)\)[/tex].
### Step 4: Plot the equations and intersection on a graph
1. Plot the first equation [tex]\(y = x + 4\)[/tex]:
- This line has a slope of 1 and y-intercept at [tex]\(y = 4\)[/tex].
- So, it will pass through points like [tex]\((0, 4)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((2, 6)\)[/tex], and so on.
2. Plot the second equation [tex]\(y = 10 - 2x\)[/tex]:
- This line has a slope of -2 and y-intercept at [tex]\(y = 10\)[/tex].
- So, it will pass through points like [tex]\((0, 10)\)[/tex], [tex]\((1, 8)\)[/tex], [tex]\((2, 6)\)[/tex], and so on.
When both lines are plotted, you will see they intersect at the point [tex]\((2, 6)\)[/tex].
### Step 5: Drawing the graph
To visualize:
1. Draw the coordinate axes.
2. Plot the line [tex]\(y = x + 4\)[/tex]. Start at the point [tex]\((0, 4)\)[/tex] and use the slope to find another point like [tex]\((1, 5)\)[/tex]. Draw the line through these points.
3. Plot the line [tex]\(y = 10 - 2x\)[/tex]. Start at the point [tex]\((0, 10)\)[/tex] and use the slope to find another point like [tex]\((1, 8)\)[/tex]. Draw the line through these points.
4. Mark the intersection point at [tex]\((2, 6)\)[/tex].
Your graph should look like this:
- A blue line (or some color) passing through [tex]\((0, 4)\)[/tex] with a slope of 1.
- A red line (or some color) passing through [tex]\( (0, 10) \)[/tex] with a slope of -2.
- They intersect at the point [tex]\((2, 6)\)[/tex].
### Step 6: Conclusion
From the graph and algebraic solution, the intersection point of the given system of linear equations is [tex]\((2, 6)\)[/tex].
This step-by-step method provides a thorough understanding of solving and graphing systems of linear equations.
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