if 5 times the first number plus 3 times the second number equals 47 and 10 times the first number minus 4 times the second number equals 54, what are the numbers?
please explain the steps
5x + 3y = 47
10x - 4y = 54
substitution:
5x = 47 - 3y, so 10x = 94-6y, thus
10x-4y = 94-6y - 4y = 54
-10y = -40
y=4
so, x = 7
elimination:
multiply the first equation by 4 and the 2nd by 3
20x+12y = 188
30x-12y = 162
add them, and the y's go away:
50x = 350
x=7
so, y=4
To solve this problem, we can use a system of equations. Let's define the first number as "x" and the second number as "y".
From the given information, we can form the following equations:
Equation 1: 5x + 3y = 47
Equation 2: 10x - 4y = 54
Now, let's solve this system of equations step by step using the method of elimination:
Step 1: Multiply Equation 1 by 2 to eliminate the coefficients of y:
(2) * (5x + 3y) = (2) * 47
10x + 6y = 94
Step 2: Multiply Equation 2 by 3 to eliminate the coefficients of x:
(3) * (10x - 4y) = (3) * 54
30x - 12y = 162
Step 3: Multiply Equation 1 by 5 and Equation 2 by 2 to make the coefficients of x the same in both equations:
(5) * (5x + 3y) = (5) * 47
(2) * (10x - 4y) = (2) * 54
25x + 15y = 235
20x - 8y = 108
Step 4: Now we have two equations with the same coefficient of x. Subtract Equation 2 from Equation 1 to eliminate x:
(25x + 15y) - (20x - 8y) = 235 - 108
25x + 15y - 20x + 8y = 127
5x + 23y = 127
Step 5: Now we have the equation 5x + 23y = 127. We can solve this equation to find the value of y.
Step 6: Subtract 5x from both sides of the equation:
5x + 23y - 5x = 127 - 5x
23y = 127 - 5x
Step 7: Divide both sides of the equation by 23:
y = (127 - 5x) / 23
Step 8: Now we substitute this value of y into Equation 1 to find the value of x:
5x + 3((127 - 5x) / 23) = 47
Step 9: Multiply through by 23 to eliminate the fraction:
23(5x) + 23(3((127 - 5x) / 23)) = 23(47)
115x + 3(127 - 5x) = 1081
Step 10: Simplify the equation:
115x + 381 - 15x = 1081
100x + 381 = 1081
Step 11: Subtract 381 from both sides of the equation:
100x + 381 - 381 = 1081 - 381
100x = 700
Step 12: Divide both sides of the equation by 100:
x = 700 / 100
x = 7
Step 13: Substitute the value of x back into Equation 1 to solve for y:
5(7) + 3y = 47
35 + 3y = 47
Step 14: Subtract 35 from both sides of the equation:
3y = 47 - 35
3y = 12
Step 15: Divide both sides of the equation by 3:
y = 12 / 3
y = 4
Therefore, the first number is 7, and the second number is 4.