y = 5x + 7
3y - 15x = 18
does this have one solution or no solutions or infinite solutions
To determine the number of solutions, we can use the method of solving a system of linear equations. We can rewrite the first equation as 5x - y = -7.
Now we have the following system of equations:
5x - y = -7 ...(1)
3y - 15x = 18 ...(2)
To solve this system, we can multiply equation (1) by 3 to make the coefficients of "y" in both equations the same:
15x - 3y = -21 ...(3)
3y - 15x = 18 ...(2)
Combining equations (2) and (3) will eliminate the "x" terms:
(3y - 15x) + (15x - 3y) = 18 + (-21)
0 = -3
Since we get an inconsistent equation (0 = -3), it means that there is no solution to this system of equations. Therefore, the original equations y = 5x + 7 and 3y - 15x = 18 do not have a solution.
To determine the number of solutions for this system of equations, we can use the concept of slope-intercept form. In slope-intercept form, an equation is written as y = mx + b, where m represents the slope of the line and b represents the y-intercept. By comparing the given equations to the slope-intercept form, we can determine the number of solutions.
Equation 1: y = 5x + 7
Equation 2: 3y - 15x = 18
1. Equation 1 is already in slope-intercept form with m = 5 and b = 7.
2. To rewrite Equation 2 in slope-intercept form, we need to isolate y. Let's rearrange Equation 2:
3y - 15x = 18
3y = 15x + 18
y = 5x + 6
Now we can compare the slopes (m) and y-intercepts (b) of the two equations:
For Equation 1:
Slope (m₁) = 5
Y-intercept (b₁) = 7
For Equation 2:
Slope (m₂) = 5
Y-intercept (b₂) = 6
Since the slopes (m₁ and m₂) are equal and the y-intercepts (b₁ and b₂) are different, the system of equations is consistent and independent, meaning it has one solution.
To determine the number of solutions for a system of linear equations, we need to put the equations in the standard form of Ax + By = C, where A, B, and C are constants.
Let's rearrange the given equations to this form:
Equation 1: y = 5x + 7
Subtracting 5x from both sides, we get: -5x + y = 7 (Equation A)
Equation 2: 3y - 15x = 18
Dividing both sides by 3, we get: -5x + y = 6 (Equation B)
Now we have two equations in the standard form. Notice that both equations are the same, -5x + y, but have different constant terms on the right side.
Since the equations represent the same line, it means that they have infinitely many solutions. Any point on this line would satisfy both equations simultaneously.
Therefore, the given system of equations has infinitely many solutions.