The difference between two numbers is 4.if the sum of five times the smaller number and the bigger number is 58,find the numbers.
smaller --- x
larger ---- x+4
5x + x+4 = 58
take over
To solve this problem, let's call the smaller number x and the bigger number y. We know that the difference between the two numbers is 4, so we can write an equation:
y - x = 4 (Equation 1)
The problem also states that the sum of five times the smaller number (5x) and the bigger number (y) is 58. We can write another equation:
5x + y = 58 (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2). We can solve it by substitution or elimination method.
Method 1: Substitution Method
From Equation 1, we can rewrite it as y = 4 + x. Now substitute this value of y into Equation 2:
5x + (4 + x) = 58
Simplify the equation:
6x + 4 = 58
Subtract 4 from both sides:
6x = 54
Divide both sides by 6:
x = 9
Now substitute the value of x back into the equation y = 4 + x:
y = 4 + 9
y = 13
So, the smaller number is 9 and the bigger number is 13.
Method 2: Elimination Method
Multiply Equation 1 by 5 to get:
5y - 5x = 20 (Equation 3)
Now subtract Equation 3 from Equation 2:
(5x + y) - (5y - 5x) = 58 - 20
Simplify the equation:
5x + y - 5y + 5x = 38
Combine like terms:
10x - 4y = 38 (Equation 4)
Now we have a new equation (Equation 4) with only x and y. We can solve it by substituting the value of x or y from Equation 1:
10(4 + x) - 4y = 38
Simplify the equation:
40 + 10x - 4y = 38
Rearrange the equation:
10x - 4y = 38 - 40
Combine like terms:
10x - 4y = -2 (Equation 5)
Now we have a new equation (Equation 5) with only x and y. We can solve it by substitution or elimination method. However, if we compare Equation 4 and Equation 5, we can see that they are the same equation. This means that the system of equations is dependent and has an infinite number of solutions. Therefore, there are multiple pairs of numbers that satisfy the given conditions.
One possible solution is x = 9 and y = 13.
In conclusion, the smaller number is 9 and the bigger number is 13.