In math class, Emma and Angelle pair as a group. Emma’s task is to graph (7/2)x−(9/2)=2y and Angelle’s task is to graph 4y=7x−9 . After they complete their graphs, they compare their work. They both get exactly the same graph. What is the solution of (7/2)x−(9/2)=2y and 4y=7x−9 ?
To find the solutions for the equations (7/2)x - (9/2) = 2y and 4y = 7x - 9, we can start by rearranging each equation so that they are in slope-intercept form (y = mx + b).
For the first equation:
(7/2)x - (9/2) = 2y
(7/2)x - (9/2) = 2y
2y = (7/2)x - (9/2)
y = (7/4)x - (9/4)
For the second equation:
4y = 7x - 9
y = (7/4)x - (9/4)
As we can see, the two equations have the exact same form, y = (7/4)x - (9/4), indicating that they represent the same line. Therefore, the solution to both equations is any point on this line.
How many solutions do these equations have
Since the two equations represent the same line, they have infinitely many solutions. Any point on the line will satisfy both equations.
To find the solution to the equations (7/2)x - (9/2) = 2y and 4y = 7x - 9, we need to find the values of x and y that satisfy both equations.
Let's start by solving the first equation: (7/2)x - (9/2) = 2y.
Step 1: Move all the terms containing y to one side of the equation by subtracting 2y from both sides:
(7/2)x - (9/2) - 2y = 0.
Step 2: Simplify the equation further by multiplying through by 2 to get rid of the fractions:
2 * [(7/2)x - (9/2) - 2y] = 2 * 0,
7x - 9 - 4y = 0.
Now, let's solve the second equation: 4y = 7x - 9.
Step 1: Move all the terms containing y to one side of the equation by subtracting 7x - 9 from both sides:
4y - (7x - 9) = 0.
Step 2: Simplify the equation further:
4y - 7x + 9 = 0.
After simplifying both equations, we now have two equations with the same form:
Equation 1: 7x - 9 - 4y = 0,
Equation 2: 4y - 7x + 9 = 0.
Now, let's compare the coefficients of x and y in both equations:
Both equations have a coefficient of -7 for x,
Both equations have a coefficient of -4 for y,
Both equations have a constant term of -9 and +9, respectively.
Since the coefficients of x and y are the same, and the constant terms are different but add up to zero, the two equations are equivalent. This means they represent the same line.
Thus, the solution to the system of equations is any point on the line given by the equation (7/2)x - (9/2) = 2y or 4y = 7x - 9.