Question: 6. A band plans to record a demo at a local studio. The cost of renting studio A is $50 plus $50 per hour. The cost of renting studio B is $100 plus $25 per hour.
The cost, $y$, in dollars of renting the studios for $x$ hours can be modelled by the linear system:
Studio A $y = 50x + 50$
Studio B $y = 25x + 100$
6a What are the $x$- and $y$-values of the intercepts of the line $y = 50x + 50$?
$x$-intercept $x = -1$
$y$-intercept $y = 50$
6b What are the $x$- and $y$-values of the intercepts of the line $y = 25x + 100$?
$x$-intercept $x =$
$y$-intercept $y =$
1. **State the problem:** We need to find the $x$- and $y$-intercepts of the lines $y = 50x + 50$ (Studio A) and $y = 25x + 100$ (Studio B).
2. **Recall intercept definitions:**
- The $y$-intercept is the value of $y$ when $x=0$.
- The $x$-intercept is the value of $x$ when $y=0$.
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### For Studio A: $y = 50x + 50$
3. **Find the $y$-intercept:**
Set $x=0$:
$$y = 50(0) + 50 = 50$$
So, the $y$-intercept is $(0, 50)$.
4. **Find the $x$-intercept:**
Set $y=0$:
$$0 = 50x + 50$$
Subtract 50 from both sides:
$$-50 = 50x$$
Divide both sides by 50:
$$x = \frac{-50}{50} = -1$$
So, the $x$-intercept is $(-1, 0)$.
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### For Studio B: $y = 25x + 100$
5. **Find the $y$-intercept:**
Set $x=0$:
$$y = 25(0) + 100 = 100$$
So, the $y$-intercept is $(0, 100)$.
6. **Find the $x$-intercept:**
Set $y=0$:
$$0 = 25x + 100$$
Subtract 100 from both sides:
$$-100 = 25x$$
Divide both sides by 25:
$$x = \frac{-100}{25} = -4$$
So, the $x$-intercept is $(-4, 0)$.
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**Final answers:**
- Studio A intercepts: $x = -1$, $y = 50$
- Studio B intercepts: $x = -4$, $y = 100$