To algebraically solve the system of equations:
1. [tex]\( y + 5x = x^2 + 10 \)[/tex]
2. [tex]\( y = 4x - 10 \)[/tex]
we proceed through the following steps:
### Step-by-Step Process:
1. Replace [tex]\( y \)[/tex] in the first equation with the expression from the second equation:
[tex]\[
y = 4x - 10
\][/tex]
Substituting into the first equation:
[tex]\[
(4x - 10) + 5x = x^2 + 10
\][/tex]
2. Simplify and combine like terms:
[tex]\[
9x - 10 = x^2 + 10
\][/tex]
3. Rearrange all terms to set the equation to zero:
[tex]\[
x^2 - 9x + 10 + 10 = 0
\][/tex]
[tex]\[
x^2 - 9x + 20 = 0
\][/tex]
4. Next, write the simplified version of the quadratic equation:
[tex]\[
x^2 - 9x = 0
\][/tex]
So, the detailed options part of the algebraically solving process include:
- Option 3: [tex]\( 0 = x^2 - 9x \)[/tex]
- Option 4: [tex]\( 0 = x^2 - 9x + 20 \)[/tex]
### Verifying One [tex]\( x \)[/tex]-value of the solution:
Given that one [tex]\( x \)[/tex]-value is 4, we can substitute [tex]\( x = 4 \)[/tex] into either equation to find [tex]\( y \)[/tex]:
Substituting [tex]\( x = 4 \)[/tex] in the second equation:
[tex]\[
y = 4(4) - 10
\][/tex]
[tex]\[
y = 16 - 10
\][/tex]
[tex]\[
y = 6
\][/tex]
Therefore, the solution [tex]\( (x, y) = (4, 6) \)[/tex] is valid.
