To solve the system of equations using substitution, we will follow these steps methodically:
Step 1: Write down the given system of equations.
[tex]\[
\begin{array}{l}
s = t + 4 \\
2t + s = 19
\end{array}
\][/tex]
Step 2: Substitute [tex]\( s \)[/tex] from the first equation into the second equation.
From equation 1, we have:
[tex]\[
s = t + 4
\][/tex]
Now substitute [tex]\( s \)[/tex] in the second equation:
[tex]\[
2t + (t + 4) = 19
\][/tex]
Step 3: Simplify and solve for [tex]\( t \)[/tex].
Combine like terms:
[tex]\[
2t + t + 4 = 19
\][/tex]
[tex]\[
3t + 4 = 19
\][/tex]
Subtract 4 from both sides to isolate the [tex]\( 3t \)[/tex] term:
[tex]\[
3t = 15
\][/tex]
Divide both sides by 3:
[tex]\[
t = 5
\][/tex]
Step 4: Substitute [tex]\( t = 5 \)[/tex] back into the first equation to find [tex]\( s \)[/tex].
Using the first equation:
[tex]\[
s = t + 4
\][/tex]
Substitute [tex]\( t = 5 \)[/tex]:
[tex]\[
s = 5 + 4
\][/tex]
[tex]\[
s = 9
\][/tex]
Step 5: Confirm the solution.
We have thus found:
[tex]\[
s = 9 \quad \text{and} \quad t = 5
\][/tex]
Step 6: Verify the values.
Substitute [tex]\( s = 9 \)[/tex] and [tex]\( t = 5 \)[/tex] back into the second equation to verify:
[tex]\[
2t + s = 19
\][/tex]
[tex]\[
2(5) + 9 = 19
\][/tex]
[tex]\[
10 + 9 = 19
\][/tex]
[tex]\[
19 = 19
\][/tex]
The solution [tex]\( (s, t) = (9, 5) \)[/tex] satisfies both equations, confirming that our solution is correct.
Step 7: Identify the correct option.
Among the provided options:
[tex]\[
(9, 5), (8, 12), (5, 9), (12, 8), (7.5, 11.5), (11.5, 7.5)
\][/tex]
The solution [tex]\( (9, 5) \)[/tex] corresponds to the first option.
Thus, the solution to the system of equations is:
[tex]\[
(s, t) = (9, 5)
\][/tex]
and the correct answer among the given options is:
[tex]\[
\text{Option 1 (9, 5)}
\][/tex]
