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Sagot :
Sure, let's solve the system of equations by graphing them step-by-step:
The system of equations is:
1. [tex]\(-3x - y = -10\)[/tex]
2. [tex]\(4x - 4y = 8\)[/tex]
### Step 1: Rewrite each equation in slope-intercept form [tex]\(y = mx + b\)[/tex].
#### Equation 1: [tex]\(-3x - y = -10\)[/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ - y = 3x - 10 \][/tex]
[tex]\[ y = -3x + 10 \][/tex]
#### Equation 2: [tex]\(4x - 4y = 8\)[/tex]
First, let's divide every term by 4 to simplify it:
[tex]\[ x - y = 2 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ -y = -x + 2 \][/tex]
[tex]\[ y = x - 2 \][/tex]
### Step 2: Graph each of the equations on the same set of axes.
#### Graph of [tex]\(y = -3x + 10\)[/tex]:
1. Find the y-intercept [tex]\((0, b)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(y = 10\)[/tex]. So, one point is [tex]\((0, 10)\)[/tex].
2. Find another point using the slope [tex]\(-3\)[/tex]: The slope is [tex]\(-3\)[/tex], meaning the rise is -3 and the run is 1. From [tex]\((0, 10)\)[/tex], move down 3 units and to the right 1 unit. This gives the point [tex]\((1, 7)\)[/tex].
#### Graph of [tex]\(y = x - 2\)[/tex]:
1. Find the y-intercept [tex]\((0, b)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex]. So, one point is [tex]\((0, -2)\)[/tex].
2. Find another point using the slope [tex]\(1\)[/tex]: The slope is [tex]\(1\)[/tex], meaning the rise is 1 and the run is 1. From [tex]\((0, -2)\)[/tex], move up 1 unit and to the right 1 unit. This gives the point [tex]\((1, -1)\)[/tex].
### Step 3: Determine the point of intersection.
When you graph both lines, you will observe that they intersect at the point where both equations satisfy the same pair of [tex]\( (x, y) \)[/tex] values.
### Step 4: Check the solution with the given values.
Since we are using the intersection point from the graph:
[tex]\[ (x, y) = (3, 1) \][/tex]
### Step 5: Verify the Point of Intersection
To verify whether [tex]\( (3, 1) \)[/tex] is the solution, we substitute these values back into the original equations:
For [tex]\( -3x - y = -10 \)[/tex]:
[tex]\[ -3(3) - 1 = -9 - 1 = -10 \quad \text{(True)} \][/tex]
For [tex]\( 4x - 4y = 8 \)[/tex]:
[tex]\[ 4(3) - 4(1) = 12 - 4 = 8 \quad \text{(True)} \][/tex]
Thus, the solution to the system of equations is indeed [tex]\( (3, 1) \)[/tex].
As a result, the point of intersection and the solution to the system is:
[tex]\[ \boxed{(3, 1)} \][/tex]
The system of equations is:
1. [tex]\(-3x - y = -10\)[/tex]
2. [tex]\(4x - 4y = 8\)[/tex]
### Step 1: Rewrite each equation in slope-intercept form [tex]\(y = mx + b\)[/tex].
#### Equation 1: [tex]\(-3x - y = -10\)[/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ - y = 3x - 10 \][/tex]
[tex]\[ y = -3x + 10 \][/tex]
#### Equation 2: [tex]\(4x - 4y = 8\)[/tex]
First, let's divide every term by 4 to simplify it:
[tex]\[ x - y = 2 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ -y = -x + 2 \][/tex]
[tex]\[ y = x - 2 \][/tex]
### Step 2: Graph each of the equations on the same set of axes.
#### Graph of [tex]\(y = -3x + 10\)[/tex]:
1. Find the y-intercept [tex]\((0, b)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(y = 10\)[/tex]. So, one point is [tex]\((0, 10)\)[/tex].
2. Find another point using the slope [tex]\(-3\)[/tex]: The slope is [tex]\(-3\)[/tex], meaning the rise is -3 and the run is 1. From [tex]\((0, 10)\)[/tex], move down 3 units and to the right 1 unit. This gives the point [tex]\((1, 7)\)[/tex].
#### Graph of [tex]\(y = x - 2\)[/tex]:
1. Find the y-intercept [tex]\((0, b)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex]. So, one point is [tex]\((0, -2)\)[/tex].
2. Find another point using the slope [tex]\(1\)[/tex]: The slope is [tex]\(1\)[/tex], meaning the rise is 1 and the run is 1. From [tex]\((0, -2)\)[/tex], move up 1 unit and to the right 1 unit. This gives the point [tex]\((1, -1)\)[/tex].
### Step 3: Determine the point of intersection.
When you graph both lines, you will observe that they intersect at the point where both equations satisfy the same pair of [tex]\( (x, y) \)[/tex] values.
### Step 4: Check the solution with the given values.
Since we are using the intersection point from the graph:
[tex]\[ (x, y) = (3, 1) \][/tex]
### Step 5: Verify the Point of Intersection
To verify whether [tex]\( (3, 1) \)[/tex] is the solution, we substitute these values back into the original equations:
For [tex]\( -3x - y = -10 \)[/tex]:
[tex]\[ -3(3) - 1 = -9 - 1 = -10 \quad \text{(True)} \][/tex]
For [tex]\( 4x - 4y = 8 \)[/tex]:
[tex]\[ 4(3) - 4(1) = 12 - 4 = 8 \quad \text{(True)} \][/tex]
Thus, the solution to the system of equations is indeed [tex]\( (3, 1) \)[/tex].
As a result, the point of intersection and the solution to the system is:
[tex]\[ \boxed{(3, 1)} \][/tex]
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