Question

find a number that is 38 less than its opposite
find a number whose product with 9 is the same as its sum with 56.
the sum of two numbers is 15. three times one of the numbers is 11 less than five times the other. find the numbers.
the difference of two integers is 9. five times the smaller is 7 more than three times the larger. find the numbers.
the lengths of the sides of a triangle are consecutive even integers. find the length of the longest side if it is 22 units shorter than the perimeter.

Explanation:

1. Find a number that is 38 less than its opposite

Step1: Let the number be \(x\)

The opposite of \(x\) is \(-x\). The equation based on the problem is \(x=-x - 38\).

Step2: Add \(x\) to both sides

\(x+x=-x - 38+x\), which simplifies to \(2x=-38\).

Step3: Solve for \(x\)

Divide both sides by 2: \(x=\frac{-38}{2}=-19\).

Step1: Let the number be \(x\)

The equation is \(9x=x + 56\).

Step2: Subtract \(x\) from both sides

\(9x-x=x + 56-x\), which gives \(8x=56\).

Step3: Solve for \(x\)

Divide both sides by 8: \(x=\frac{56}{8}=7\).

Step1: Let the two numbers be \(x\) and \(y\)

We have the system of equations \(

$$\begin{cases}x + y=15\\3x=5y-11\end{cases}$$

\)
From the first - equation \(x = 15 - y\).

Step2: Substitute \(x = 15 - y\) into the second equation

\(3(15 - y)=5y-11\).
Expand the left - hand side: \(45-3y=5y-11\).

Step3: Add \(3y\) to both sides

\(45-3y + 3y=5y-11+3y\), which gives \(45 = 8y-11\).

Step4: Add 11 to both sides

\(45 + 11=8y-11+11\), so \(56 = 8y\).

Step5: Solve for \(y\)

Divide both sides by 8: \(y = 7\).

Step6: Find \(x\)

Substitute \(y = 7\) into \(x=15 - y\), then \(x=15 - 7 = 8\).

Answer:

\(-19\)

2. Find a number whose product with 9 is the same as its sum with 56