Answer :
We start by solving the first pair of equations:
[tex]$$
T = 150c - 3 \quad \text{and} \quad T = -3c + 150.
$$[/tex]
Step 1: Set the two expressions for [tex]$T$[/tex] equal to each other:
[tex]$$
150c - 3 = -3c + 150.
$$[/tex]
Step 2: Add [tex]$3c$[/tex] to both sides to combine like terms:
[tex]$$
150c + 3c - 3 = 150,
$$[/tex]
which simplifies to
[tex]$$
153c - 3 = 150.
$$[/tex]
Step 3: Add [tex]$3$[/tex] to both sides:
[tex]$$
153c = 153.
$$[/tex]
Step 4: Divide both sides by [tex]$153$[/tex]:
[tex]$$
c = \frac{153}{153} = 1.
$$[/tex]
Step 5: Substitute [tex]$c = 1$[/tex] into one of the original equations to find [tex]$T$[/tex]. Using [tex]$T = 150c - 3$[/tex]:
[tex]$$
T = 150(1) - 3 = 147.
$$[/tex]
So, the solution for the first system is [tex]$c = 1$[/tex] and [tex]$T = 147$[/tex].
Now, let’s solve the second pair of equations:
[tex]$$
T = -150c + 3 \quad \text{and} \quad T = 3c - 150.
$$[/tex]
Step 1: Set the two expressions for [tex]$T$[/tex] equal to each other:
[tex]$$
-150c + 3 = 3c - 150.
$$[/tex]
Step 2: Add [tex]$150c$[/tex] to both sides to combine like terms:
[tex]$$
3 = 3c + 150c - 150,
$$[/tex]
which simplifies to
[tex]$$
3 = 153c - 150.
$$[/tex]
Step 3: Add [tex]$150$[/tex] to both sides:
[tex]$$
153 = 153c.
$$[/tex]
Step 4: Divide both sides by [tex]$153$[/tex]:
[tex]$$
c = \frac{153}{153} = 1.
$$[/tex]
Step 5: Substitute [tex]$c = 1$[/tex] into one of the equations to find [tex]$T$[/tex]. Using [tex]$T = -150c + 3$[/tex]:
[tex]$$
T = -150(1) + 3 = -150 + 3 = -147.
$$[/tex]
Thus, the solution for the second system is [tex]$c = 1$[/tex] and [tex]$T = -147$[/tex].
In summary, the solutions are:
For the first system: [tex]$c = 1$[/tex], [tex]$T = 147$[/tex], and for the second system: [tex]$c = 1$[/tex], [tex]$T = -147$[/tex].
[tex]$$
T = 150c - 3 \quad \text{and} \quad T = -3c + 150.
$$[/tex]
Step 1: Set the two expressions for [tex]$T$[/tex] equal to each other:
[tex]$$
150c - 3 = -3c + 150.
$$[/tex]
Step 2: Add [tex]$3c$[/tex] to both sides to combine like terms:
[tex]$$
150c + 3c - 3 = 150,
$$[/tex]
which simplifies to
[tex]$$
153c - 3 = 150.
$$[/tex]
Step 3: Add [tex]$3$[/tex] to both sides:
[tex]$$
153c = 153.
$$[/tex]
Step 4: Divide both sides by [tex]$153$[/tex]:
[tex]$$
c = \frac{153}{153} = 1.
$$[/tex]
Step 5: Substitute [tex]$c = 1$[/tex] into one of the original equations to find [tex]$T$[/tex]. Using [tex]$T = 150c - 3$[/tex]:
[tex]$$
T = 150(1) - 3 = 147.
$$[/tex]
So, the solution for the first system is [tex]$c = 1$[/tex] and [tex]$T = 147$[/tex].
Now, let’s solve the second pair of equations:
[tex]$$
T = -150c + 3 \quad \text{and} \quad T = 3c - 150.
$$[/tex]
Step 1: Set the two expressions for [tex]$T$[/tex] equal to each other:
[tex]$$
-150c + 3 = 3c - 150.
$$[/tex]
Step 2: Add [tex]$150c$[/tex] to both sides to combine like terms:
[tex]$$
3 = 3c + 150c - 150,
$$[/tex]
which simplifies to
[tex]$$
3 = 153c - 150.
$$[/tex]
Step 3: Add [tex]$150$[/tex] to both sides:
[tex]$$
153 = 153c.
$$[/tex]
Step 4: Divide both sides by [tex]$153$[/tex]:
[tex]$$
c = \frac{153}{153} = 1.
$$[/tex]
Step 5: Substitute [tex]$c = 1$[/tex] into one of the equations to find [tex]$T$[/tex]. Using [tex]$T = -150c + 3$[/tex]:
[tex]$$
T = -150(1) + 3 = -150 + 3 = -147.
$$[/tex]
Thus, the solution for the second system is [tex]$c = 1$[/tex] and [tex]$T = -147$[/tex].
In summary, the solutions are:
For the first system: [tex]$c = 1$[/tex], [tex]$T = 147$[/tex], and for the second system: [tex]$c = 1$[/tex], [tex]$T = -147$[/tex].
