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Sagot :
To find the solution to the system of equations, we will graph both equations and identify the point of intersection. The intersection point, if there is one, will be the solution to the system. Here are the steps:
### Step 1: Graph the First Equation
The first equation is:
[tex]\[ y = -\frac{1}{4} x - 3 \][/tex]
This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = -\frac{1}{4} \)[/tex] is the slope, and [tex]\( b = -3 \)[/tex] is the y-intercept.
- The y-intercept is the point [tex]\((0, -3)\)[/tex].
- The slope [tex]\(-\frac{1}{4}\)[/tex] means that for every increase of 4 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1 unit.
To get another point, start from the y-intercept:
- Increase [tex]\( x \)[/tex] by 4 (from 0 to 4), and decrease [tex]\( y \)[/tex] by 1 (from -3 to -4).
- This gives us another point: (4, -4).
Plot these points on the graph and draw the line through them.
### Step 2: Graph the Second Equation
The second equation is:
[tex]\[ -2x + y = 6 \][/tex]
To put this equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = 2x + 6 \][/tex]
This is a linear equation where the slope [tex]\( m = 2 \)[/tex] and the y-intercept [tex]\( b = 6 \)[/tex].
- The y-intercept is the point [tex]\((0, 6)\)[/tex].
- The slope [tex]\( 2 \)[/tex] means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
To get another point, start from the y-intercept:
- Increase [tex]\( x \)[/tex] by 1 (from 0 to 1), and increase [tex]\( y \)[/tex] by 2 (from 6 to 8).
- This gives us another point: (1, 8).
Plot these points on the graph and draw the line through them.
### Step 3: Find the Intersection Point
To find the intersection point, set the two equations equal to solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
From the first equation:
[tex]\[ y = -\frac{1}{4} x - 3 \][/tex]
From the second equation:
[tex]\[ y = 2x + 6 \][/tex]
Set them equal to each other:
[tex]\[ -\frac{1}{4} x - 3 = 2x + 6 \][/tex]
Solve for [tex]\( x \)[/tex]:
1. Multiply both sides by 4 to eliminate the fraction:
[tex]\[ -x - 12 = 8x + 24 \][/tex]
2. Combine like terms:
[tex]\[ -x - 8x = 24 + 12 \][/tex]
[tex]\[ -9x = 36 \][/tex]
3. Divide by -9:
[tex]\[ x = -4 \][/tex]
Once we have [tex]\( x \)[/tex], we substitute it back into one of the original equations to find [tex]\( y \)[/tex]. Substitute [tex]\( x = -4 \)[/tex] into [tex]\( y = 2x + 6 \)[/tex]:
[tex]\[ y = 2(-4) + 6 \][/tex]
[tex]\[ y = -8 + 6 \][/tex]
[tex]\[ y = -2 \][/tex]
### Step 4: Conclusion
The intersection point, and therefore the solution to the system of equations, is [tex]\( (-4, -2) \)[/tex].
### Graph Verification
1. For the first equation [tex]\( y = -\frac{1}{4}x - 3 \)[/tex]:
- At [tex]\( x = -4 \)[/tex]: [tex]\( y = -\frac{1}{4}(-4) - 3 = 1 - 3 = -2 \)[/tex]
2. For the second equation [tex]\( y = 2x + 6 \)[/tex]:
- At [tex]\( x = -4 \)[/tex]: [tex]\( y = 2(-4) + 6 = -8 + 6 = -2 \)[/tex]
Both graphs intersect at the point [tex]\( (-4, -2) \)[/tex], confirming that the solution to the system of equations is:
[tex]\[ (-4, -2) \][/tex]
### Step 1: Graph the First Equation
The first equation is:
[tex]\[ y = -\frac{1}{4} x - 3 \][/tex]
This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = -\frac{1}{4} \)[/tex] is the slope, and [tex]\( b = -3 \)[/tex] is the y-intercept.
- The y-intercept is the point [tex]\((0, -3)\)[/tex].
- The slope [tex]\(-\frac{1}{4}\)[/tex] means that for every increase of 4 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1 unit.
To get another point, start from the y-intercept:
- Increase [tex]\( x \)[/tex] by 4 (from 0 to 4), and decrease [tex]\( y \)[/tex] by 1 (from -3 to -4).
- This gives us another point: (4, -4).
Plot these points on the graph and draw the line through them.
### Step 2: Graph the Second Equation
The second equation is:
[tex]\[ -2x + y = 6 \][/tex]
To put this equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = 2x + 6 \][/tex]
This is a linear equation where the slope [tex]\( m = 2 \)[/tex] and the y-intercept [tex]\( b = 6 \)[/tex].
- The y-intercept is the point [tex]\((0, 6)\)[/tex].
- The slope [tex]\( 2 \)[/tex] means that for every increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
To get another point, start from the y-intercept:
- Increase [tex]\( x \)[/tex] by 1 (from 0 to 1), and increase [tex]\( y \)[/tex] by 2 (from 6 to 8).
- This gives us another point: (1, 8).
Plot these points on the graph and draw the line through them.
### Step 3: Find the Intersection Point
To find the intersection point, set the two equations equal to solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
From the first equation:
[tex]\[ y = -\frac{1}{4} x - 3 \][/tex]
From the second equation:
[tex]\[ y = 2x + 6 \][/tex]
Set them equal to each other:
[tex]\[ -\frac{1}{4} x - 3 = 2x + 6 \][/tex]
Solve for [tex]\( x \)[/tex]:
1. Multiply both sides by 4 to eliminate the fraction:
[tex]\[ -x - 12 = 8x + 24 \][/tex]
2. Combine like terms:
[tex]\[ -x - 8x = 24 + 12 \][/tex]
[tex]\[ -9x = 36 \][/tex]
3. Divide by -9:
[tex]\[ x = -4 \][/tex]
Once we have [tex]\( x \)[/tex], we substitute it back into one of the original equations to find [tex]\( y \)[/tex]. Substitute [tex]\( x = -4 \)[/tex] into [tex]\( y = 2x + 6 \)[/tex]:
[tex]\[ y = 2(-4) + 6 \][/tex]
[tex]\[ y = -8 + 6 \][/tex]
[tex]\[ y = -2 \][/tex]
### Step 4: Conclusion
The intersection point, and therefore the solution to the system of equations, is [tex]\( (-4, -2) \)[/tex].
### Graph Verification
1. For the first equation [tex]\( y = -\frac{1}{4}x - 3 \)[/tex]:
- At [tex]\( x = -4 \)[/tex]: [tex]\( y = -\frac{1}{4}(-4) - 3 = 1 - 3 = -2 \)[/tex]
2. For the second equation [tex]\( y = 2x + 6 \)[/tex]:
- At [tex]\( x = -4 \)[/tex]: [tex]\( y = 2(-4) + 6 = -8 + 6 = -2 \)[/tex]
Both graphs intersect at the point [tex]\( (-4, -2) \)[/tex], confirming that the solution to the system of equations is:
[tex]\[ (-4, -2) \][/tex]
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