Let's solve the questions step-by-step:
Question 1:
Let \( s \) be the number of student tickets and \( a \) be the number of adult tickets. We have the following system of equations:
- \( s + a = 400 \) (total tickets)
- \( 6s + 11a = 3000 \) (total sales)
From the first equation, we can express \( a \) in terms of \( s \): \[ a = 400 - s \]
Substituting this into the second equation: \[ 6s + 11(400 - s) = 3000 \] \[ 6s + 4400 - 11s = 3000 \] \[ -5s + 4400 = 3000 \] \[ -5s = 3000 - 4400 \] \[ -5s = -1400 \] \[ s = 280 \]
Answer: 280
Question 2:
Let \( h \) be the cost of one hot dog and \( w \) be the cost of one bottle of water. We can create the following system from the two families' purchases:
- \( 4h + 6w = 22 \)
- \( 8h + 3w = 35 \)
We can solve this system using elimination or substitution. Let’s first simplify the first equation: \[ 2h + 3w = 11 \] \( 8h + 3w = 35 \) remains the same.
Now, subtract the first new equation from the second: \[ (8h + 3w) - (2h + 3w) = 35 - 11 \] \[ 6h = 24 \] \[ h = 4 \]
Answer: $4
Question 3:
The graph showing a system of equations with no solution is one where the two lines are parallel, meaning they never intersect.
Answer: The graph shows two parallel lines.
Question 4:
To solve the system of equations using substitution:
- \( 3x + 2y = 7 \)
- \( x = 3y + 6 \)
Substituting \( x \) from the second equation into the first: \[ 3(3y + 6) + 2y = 7 \] \[ 9y + 18 + 2y = 7 \] \[ 11y + 18 = 7 \] \[ 11y = 7 - 18 \] \[ 11y = -11 \] \[ y = -1 \]
Now substitute \( y \) back into \( x = 3y + 6 \): \[ x = 3(-1) + 6 = 3 \]
Answer: (3, −1)
Question 5:
The inequality \( 3x - 2y > 4 \) results in a dashed line because the inequality is strict (">"). To find its intercepts:
- For y-intercept (\( x = 0 \)): \[ -2y = 4 \Rightarrow y = -2 \]
- For x-intercept (\( y = 0 \)): \[ 3x = 4 \Rightarrow x = \frac{4}{3} \]
The graph has shading above the line since it is "greater than".
Answer: The graph shows a dashed line, which crosses the y-axis at negative 2 and the x-axis at 4 thirds, with shading above the line.
Question 6:
We need to check which ordered pair satisfies the inequalities:
- \( y ≥ \frac{2}{3}x + 2 \)
- \( y < -\frac{1}{3}x + 1 \)
Checking point (−6, −3):
- \( -3 ≥ \frac{2}{3}(-6) + 2 \rightarrow -3 ≥ -4 + 2 \rightarrow -3 ≥ -2 \) (not true)
- \( -3 < -\frac{1}{3}(-6) + 1 \rightarrow -3 < 2 + 1 \rightarrow -3 < 3 \) (true)
(−6, −2):
- \( -2 ≥ -4 + 2 \rightarrow -2 ≥ -2 \) (not true)
- \( -2 < 2 + 1 \rightarrow -2 < 3 \) (true)
(−6, 3):
- \( 3 ≥ -4 + 2 \rightarrow 3 ≥ -2 \) (true)
- \( 3 < 3 \) (not true)
(−6, 4):
- \( 4 ≥ -4 + 2 \rightarrow 4 ≥ -2 \) (true)
- \( 4 < 3 \) (not true)
Answer: (−6, −3)
Question 7:
Let \( t \) be the number of t-shirts and \( y \) be the number of yearbooks. The inequality representing the sales requirement is: \[ 22t + 23y ≥ 2400 \]
Let's check each option:
- \( 50(22) + 56(23) = 1100 + 1288 = 2388 \) (not sufficient)
- \( 51(22) + 55(23) = 1122 + 1265 = 2387 \) (not sufficient)
- \( 52(22) + 54(23) = 1144 + 1242 = 2386 \) (not sufficient)
- \( 53(22) + 55(23) = 1166 + 1265 = 2431 \) (sufficient)
Answer: 53 t-shirts and 55 yearbooks
Question 8:
The equations can be solved for their slopes and intercepts to find how they intersect.
-
First equation: \(-2x - y = 10 \rightarrow y = -2x - 10\) (slope = -2)
-
Second equation: \(x - 2y = 5 \rightarrow 2y = x - 5 \rightarrow y = \frac{1}{2}x + \frac{5}{2}\) (slope = \(\frac{1}{2}\))
Since their slopes are different, they will intersect at a single point.
Answer: The lines intersect at 3 comma 4.
Question 9:
To eliminate \( y \), we can manipulate each equation:
-
Multiply the second equation by 2: \[ 6x - 8y = -36 \]
-
Now we have: \[ 6x + 2y = -6 \] \[ 6x - 8y = -36 \]
Subtract the first from the second: \[ (6x - 8y) - (6x + 2y) = -36 - (-6) \] \[ -10y = -30 \] \[ y = 3 \]
Substituting back into \( 6x + 2(3)= -6 \) yields: \[ 6x + 6 = -6 \rightarrow 6x = -12 \rightarrow x = -2 \]
Answer: (−2, 3)
Question 10:
Calculating each option:
- \( 20(15) + 65(7) = 300 + 455 = 755\) (not sufficient)
- \( 25(15) + 60(7) = 375 + 420 = 795\) (not sufficient)
- \( 26(15) + 61(7) = 390 + 427 = 817\) (sufficient)
- \( 14(15) + 45(7) = 210 + 315 = 525\) (not sufficient)
Answer: 26 paintings and 61 bracelets
Question 11:
From the equations:
- \( x = 5y \)
- \( x = 8 + 3y \)
Setting them equal gives: \[ 5y = 8 + 3y \] \[ 2y = 8 \] \[ y = 4 \] So, \( x = 5(4) = 20 \)
Answer: Nikhil has been with the company for 20 years, while Mae has been there for 4 years.
Question 12:
Finding intersections by solving:
- For \( y = 3x + 9 \)
- For \( 6x - 2y = 6 \) rearranging gives \( y = 3x - 3 \)
Setting \( 3x + 9 = 3x - 3 \): This results in no solution as they are the same slope.
Answer: There is no solution.
Question 13:
The inequality \( y ≥ -\frac{1}{3} x + 1 \) means the line is solid and shading is above.
Answer: The graph shows a solid line, which crosses the y-axis at 1 and the x-axis at 3, with shading above the line.
Question 14:
Using elimination for the system:
- \( -2x + 3y = 13 \)
- \( x + y = 11 \)
From the second equation, solve for \( y \): \[ y = 11 - x\]
Substituting into the first equation: \[ -2x + 3(11 - x) = 13 \] \[ -2x + 33 - 3x = 13 \] \[ -5x + 33 = 13 \] \[ -5x = -20 \] \[ x = 4 \] Substituting back gives \( y = 7 \).
Answer: (4, 7)
Question 15:
To determine from the inequalities:
- \( y ≤ -5x + 2\) is solid and below.
- \( y < 3x - 1.5\) is a dashed line and above.
Answer: The graph shows a solid line that passes through (0, 2) and (1, -3), with shading below the line. There is also a dashed line that passes through (0, -1.5) and (1, 1.5), with shading above the line.
If you have further questions or topics, feel free to ask!
2 answers