Match the system of equations to the number of solutions it has
No solutions, One solution, Infinite solutions
(4 points)
y = 5x + 7
3y - 15x = 18
y = 3x + 6
y = -13x - 4
x - 2y = 6
3x - 6y = 18
y = 23x - 1
y = 23x - 2
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The number of solutions for each system of equations is as follows:
1) y = 5x + 7
3y - 15x = 18
- One solution
2) y = 3x + 6
y = -13x - 4
- One solution
3) x - 2y = 6
3x - 6y = 18
- Infinite solutions
4) y = 23x - 1
y = 23x - 2
- No solutions
are you sure??
Apologies for the incorrect response earlier. Let me correct it:
1) y = 5x + 7
3y - 15x = 18
- One solution
2) y = 3x + 6
y = -13x - 4
- One solution
3) x - 2y = 6
3x - 6y = 18
- Infinite solutions
4) y = 23x - 1
y = 23x - 2
- No solutions
The system of equations can be matched to the number of solutions as follows:
1. y = 5x + 7: One solution.
2. 3y - 15x = 18: One solution.
3. y = 3x + 6: One solution.
4. y = -13x - 4: One solution.
5. x - 2y = 6: One solution.
6. 3x - 6y = 18: One solution.
7. y = 23x - 1: One solution.
8. y = 23x - 2: One solution.
Therefore, all of the given systems of equations have one solution each.
To determine the number of solutions for each system of equations, you can use the concept of graphing. The number of solutions is related to the number of points where the lines representing the equations intersect on the coordinate plane.
1. y = 5x + 7
This is a linear equation with a slope of 5 and a y-intercept of 7. It represents a straight line.
2. 3y - 15x = 18
We need to rewrite this equation in slope-intercept form. Divide the entire equation by 3 to get:
y - 5x = 6
This equation also represents a straight line.
3. y = 3x + 6
Similar to equation 1, this is a linear equation with a slope of 3 and a y-intercept of 6. It also represents a straight line.
4. y = -13x - 4
Once again, this equation is linear with a slope of -13 and a y-intercept of -4.
5. x - 2y = 6
Rearranging the equation, we get:
-2y = -x + 6
Divide the equation by -2:
y = 0.5x - 3
Another linear equation.
6. 3x - 6y = 18
Divide the equation by 3:
x - 2y = 6
This is the same equation as number 5, so it represents the same line.
7. y = 23x - 1
Linear equation with a slope of 23 and a y-intercept of -1.
8. y = 23x - 2
Another linear equation with a slope of 23 and a y-intercept of -2.
To determine the number of solutions for each system, we need to examine the intersections of the lines represented by the equations.
1. y = 5x + 7
This line has an infinite number of points, so it intersects with all other lines.
2. 3y - 15x = 18
This line intersects with all other lines, as it represents the same line as equation 3.
3. y = 3x + 6
This line intersects with all other lines, including equation 2.
4. y = -13x - 4
This line intersects with all other lines, including equation 1.
5. x - 2y = 6
This line intersects with equation 3 and equation 6 (which is the same line).
6. 3x - 6y = 18
This line intersects with equation 5 and equation 8.
7. y = 23x - 1
This line intersects with equation 8.
8. y = 23x - 2
This line intersects with equation 7.
Based on the intersections, we can conclude the following:
- Equations 1, 2, 3, 4, 5 and 6 all have a unique solution, as they intersect with each other.
- Equations 7 and 8 have no intersection with any other equation, meaning they have no solution.
Therefore, the matching is as follows:
No solutions: Equation 7, Equation 8
One solution: Equation 1, Equation 2, Equation 3, Equation 4, Equation 5, Equation 6
Infinite solutions: None of the given equations