The difference of two integers is 9. Five times the smaller is 7 more than three times the larger. Find the numbers.
Let's assume the smaller integer as 'x' and the larger integer as 'y'.
1. From the given information, we can write the first equation as:
y - x = 9 (Equation 1)
2. The second equation states that five times the smaller integer is 7 more than three times the larger integer:
5x = 3y + 7 (Equation 2)
To solve the system of equations, we will use the substitution method.
Step 1: Solve Equation 1 for y
y = x + 9
Step 2: Substitute the value of y in Equation 2 with x + 9
5x = 3(x + 9) + 7
Simplifying the equation:
5x = 3x + 27 + 7
5x = 3x + 34
Step 3: Subtract 3x from both sides of the equation
5x - 3x = 3x - 3x + 34
2x = 34
Step 4: Divide both sides of the equation by 2
x = 34/2
x = 17
Step 5: Substitute the value of x back into Equation 1 to find y
y = x + 9
y = 17 + 9
y = 26
Therefore, the smaller integer is 17 and the larger integer is 26.
To find the numbers, let's assign variables.
Let's say the smaller number is x, and the larger number is y. According to the problem, we know two things:
1) The difference of two integers is 9:
This can be written as an equation: y - x = 9.
2) Five times the smaller is 7 more than three times the larger:
This can be written as an equation: 5x = 3y + 7.
Now, we have a system of two equations with two variables. We can solve this system using either substitution or elimination method. Let's use the substitution method.
Step 1: Solve the first equation for y in terms of x:
y - x = 9
y = x + 9
Step 2: Substitute the value of y in the second equation:
5x = 3(x + 9) + 7
Step 3: Simplify the equation:
5x = 3x + 27 + 7
5x = 3x + 34
Step 4: Move all the variables to one side of the equation:
5x - 3x = 34
2x = 34
Step 5: Solve for x:
x = 34 / 2
x = 17
Step 6: Substitute the value of x back into either of the original equations to find y.
y = x + 9
y = 17 + 9
y = 26
Therefore, the two numbers are 17 and 26.