Answer:
The two numbers are 9 and -26.
The larger number is 9.
Explanation:
Let the two unknown numbers be x and y.
The sum of a four times a number x and another number y is 10. Hence:
![4x+y=10](https://img.qammunity.org/2021/formulas/mathematics/middle-school/y5l80s0sgpyird82okwzo0ijhzyfv1ar9l.png)
And the difference between the two numbers is 35. So:
![x-y=35](https://img.qammunity.org/2021/formulas/mathematics/high-school/mw2hbjhdzpy0hfho3d3clu498g1u9gqw4a.png)
So, we have the system of equations:
![\left\{ \begin{array} \ 4x+y=10 \\ x-y=35 \end{array}](https://img.qammunity.org/2021/formulas/mathematics/high-school/sxqw6h1ku3lhemndbunu5355vh1m3maeil.png)
From the second equation, we can add y to both sides:
![x=y+35](https://img.qammunity.org/2021/formulas/mathematics/high-school/n94pg83uas90byoxgudg0qrbtlyu1i0cbm.png)
Now, we can substitute this into the first equation:
![4(y+35)+y=10](https://img.qammunity.org/2021/formulas/mathematics/high-school/73ef92pal6s9lhfnjefpglxsjtg3hmy3kz.png)
Distribute:
![4y+140+y=10](https://img.qammunity.org/2021/formulas/mathematics/high-school/vpr0mhrraly4a2gad6kq99lobxxm6m7avc.png)
Simplify:
![5y+140=10](https://img.qammunity.org/2021/formulas/mathematics/high-school/y0lcgggpwyxt1vt1jabfgjq3n4nca8bdyu.png)
Subtract 140 from both sides:
![5y=-130](https://img.qammunity.org/2021/formulas/mathematics/high-school/td4hehwwer5beynylb2a0s3fda4c4ml40k.png)
Divide both sides by 5:
![y=-26](https://img.qammunity.org/2021/formulas/mathematics/high-school/nyp5uf2rv6qfxk1htszqypgkxo3no4jgmy.png)
So, the value of y is -26 .
Then it follows that the value of x is:
![x=y+35](https://img.qammunity.org/2021/formulas/mathematics/high-school/n94pg83uas90byoxgudg0qrbtlyu1i0cbm.png)
Substitute -26 for y:
![x=-26+35=9](https://img.qammunity.org/2021/formulas/mathematics/high-school/cdx9o8ykon18wtcplhaieun8csk4ty428x.png)
So, our two numbers are 9 and -26.
The larger number is 9.
So, our answer is 9.