Final Answer:
Using the substitution method in simultaneous equations, the solution is x = 3/5 and y = -3/4
Step-by-step explanation:
To solve the system of simultaneous equations:
1) 15x - 7y = 14 1/4
2) 5x - y = 3 3/4
using the substitution method, follow these steps:
Step 1: Convert the fractions in the equations to improper forms.
14 1/4 = (14*4 + 1)/4 = 57/4
3 3/4 = (3*4 + 3)/4 = 15/4
Now our equations look like:
1) 15x - 7y = 57/4
2) 5x - y = 15/4
Step 2: Look for the easiest equation to express one variable in terms of the other. In this case, it is easier to express 'y' from the second equation.
5x - y = 15/4
Add 'y' to both sides and subtract (15/4) from both sides:
y = 5x - 15/4
Now we have 'y' in terms of 'x'.
Step 3: Substitute 'y' in the first equation with the expression we found from the second equation.
15x - 7(5x - 15/4) = 57/4
Step 4: Distribute -7 into the parentheses of the expression:
15x - 35x + (7*15)/4 = 57/4
Simplify further:
-20x + 105/4 = 57/4
Step 5: Isolate the variable 'x' by moving the fraction to the other side:
-20x = 57/4 - 105/4
Step 6: Combine the fractions on the right side:
-20x = (57 - 105)/4
-20x = -48/4
Reduce the fraction by dividing both numerator and denominator by 4:
-20x = -12
Step 7: Solve for 'x' by dividing both sides by -20:
x = -12 / -20
Simplify the fraction by dividing top and bottom by 4:
x = 3/5
Step 8: Now we can substitute the value of 'x' back into the equation we found for 'y':
y = 5x - 15/4
y = 5(3/5) - 15/4
The 5's cancel out in the first term:
y = 3 - 15/4
Since we have a mixed number, express 3 as a fraction with a denominator of 4:
y = 12/4 - 15/4
Combine the fractions:
y = (12 - 15)/4
y = -3/4
We have now found the values for both 'x' and 'y':
x = 3/5
y = -3/4
These are the solutions to the system of equations using the substitution method.