Question

linear equations and inequalities
the questions in the first section on a test are worth 3 points and the questions in the second section are worth 5 points. let x represent the number of correct questions from the first section, and let y represent the number of correct questions from the second section. sydney earned more than 60 points on the test. the inequality representing her score is 3x + 5y > 60.

if sydney answered 15 questions in the first section correctly, what is the minimum number of questions she must have answered correctly in the second section?

options: 8, 2, 1, 7

Explanation:

Step1: Substitute \( x = 15 \) into the inequality

We know the inequality is \( 3x + 5y>60 \), and \( x = 15 \). Substitute \( x \) into the inequality: \( 3\times15 + 5y>60 \). Calculate \( 3\times15 = 45 \), so the inequality becomes \( 45+5y>60 \).

Step2: Solve for \( y \)

Subtract 45 from both sides of the inequality: \( 5y>60 - 45 \). Calculate \( 60 - 45 = 15 \), so \( 5y>15 \). Divide both sides by 5: \( y>\frac{15}{5}=3 \). Since \( y \) represents the number of correct questions (a non - negative integer), the minimum integer value of \( y \) that satisfies \( y > 3 \) is 4? Wait, there is a mistake above. Wait, let's recalculate:

Wait, the original inequality: \( 3x+5y > 60 \), \( x = 15 \), so \( 3\times15=45 \), then \( 45 + 5y>60 \), subtract 45: \( 5y>15 \), \( y > 3 \). But the options are 8,2,1,7. Wait, maybe I misread the inequality. Wait, maybe the inequality is \( 3x + 5y\geq60 \)? Or maybe the total points are different? Wait, let's check the options. If \( y = 7 \), then \( 3\times15+5\times7=45 + 35=80>60 \). If \( y = 4 \), but 4 is not in the options. Wait, maybe the inequality is \( 3x+5y>60 \), and we made a mistake. Wait, let's check the options again. The options are 8,2,1,7.

Wait, maybe the inequality is \( 3x + 5y>60 \), \( x = 15 \), so \( 3\times15 = 45 \), so \( 5y>60 - 45=15 \), \( y > 3 \). The smallest integer \( y \) in the options that is greater than 3 is 7? Wait, no, 8 is also greater. Wait, maybe the original problem has a different total. Wait, maybe the inequality is \( 3x+5y>60 \), and we need to find the minimum \( y \) such that \( 3\times15 + 5y>60 \). Let's check each option:

• For \( y = 1 \): \( 45+5\times1=50<60 \)
• For \( y = 2 \): \( 45 + 10=55<60 \)
• For \( y = 7 \): \( 45+35 = 80>60 \)
• For \( y = 8 \): \( 45+40 = 85>60 \)

Since we need the minimum \( y \) that satisfies the inequality, among the given options, 7 is the smallest one that satisfies \( y>3 \) (because 7>3, and 2,1 are less than or equal to 3, 8 is also greater but 7 is smaller than 8 among the valid ones). Wait, but 7 is in the options. So the minimum number of questions is 7? Wait, but let's re - check the calculation.

Wait, \( 3x+5y>60 \), \( x = 15 \), so \( 3\times15=45 \), so \( 5y>15 \), \( y > 3 \). The integers \( y \) greater than 3 are 4,5,6,7,8,... Among the options (1,2,7,8), the smallest one is 7. So the answer is 7.

Answer:

7