Sure, let's solve this step by step.
1. Let's denote the smaller number as [tex]\( x \)[/tex] and the larger number as [tex]\( y \)[/tex].
2. According to the problem, we have the first condition:
[tex]\[
y = x + 6
\][/tex]
This means the larger number [tex]\( y \)[/tex] is 6 more than the smaller number [tex]\( x \)[/tex].
3. The second condition given in the problem is:
[tex]\[
7x = 10y
\][/tex]
This means 7 times the smaller number [tex]\( x \)[/tex] equals 10 times the larger number [tex]\( y \)[/tex].
4. Now we can use these two equations to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
5. From the first equation, we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = x + 6
\][/tex]
6. Substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[
7x = 10(x + 6)
\][/tex]
7. Now, simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
7x = 10x + 60
\][/tex]
[tex]\[
7x - 10x = 60
\][/tex]
[tex]\[
-3x = 60
\][/tex]
[tex]\[
x = -20
\][/tex]
8. Now that we have [tex]\( x \)[/tex], we can find [tex]\( y \)[/tex] using the first equation:
[tex]\[
y = x + 6
\][/tex]
[tex]\[
y = -20 + 6
\][/tex]
[tex]\[
y = -14
\][/tex]
So, the two numbers are:
- The smaller number [tex]\( x \)[/tex] is -20
- The larger number [tex]\( y \)[/tex] is -14
