Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Two classmates got together over the weekend to do their assigned History reading. Sophie can read 1 page per minute, while Quincy can read 4 pages per minute. When they met, Sophie had already read 83 pages, and Quincy had already gotten through 11 pages. After a while, they had both read the same number of pages. How many pages had each one read? How long did that take? Sophie and Quincy had each read _____ pages after _____ minutes.

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.  Two classmates got together over the weekend to do their assigned History reading. Sophie can read 1 page per minute, while Quincy can read 4 pages per minute. When they met, Sophie had already read 83 pages, and Quincy had already gotten through 11 pages. After a while, they had both read the same number of pages. How many pages had each one read? How long did that take?   Sophie and Quincy had each read _____ pages after _____ minutes.

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Define Variables: Let's define the variables for the system of equations. Let x represent the number of minutes they read together, and let S represent the total number of pages Sophie has read, and Q represent the total number of pages Quincy has read. Sophie's rate is 1 page per minute, and she had already read 83 pages. So, her equation will be: S = 1 * x + 83 Quincy's rate is 4 pages per minute, and he had already read 11 pages. So, his equation will be: Q = 4 * x + 11 We are given that after a while, they had both read the same number of pages, which means S = Q.Set Up Equations: Now we set up the system of equations using the information we have: 1. S = x + 83 2. Q = 4x + 11 And since S = Q, we can write: x + 83 = 4x + 11Solve for x: Next, we solve for x using substitution. We already have S = Q, so we can substitute S for Q in the second equation: x + 83 = 4x + 11 Now, we solve for x by getting all the x terms on one side and the constants on the other: x - 4x = 11 - 83 -3x = -72Find Total Time: Divide both sides by -3 to find the value of x: -3x / -3 = -72 / -3 x = 24 This means they read together for 24 minutes.Find Sophie's Pages: Now that we have the value of x, we can find out how many pages each person read. We'll substitute x back into one of the original equations. Let's use Sophie's equation: S = x + 83 S = 24 + 83 S = 107 So, Sophie read 107 pages in total.Check Quincy's Pages: We can also check Quincy's total pages read to ensure our solution is consistent: Q = 4x + 11 Q = 4 * 24 + 11 Q = 96 + 11 Q = 107 Quincy also read 107 pages in total, which confirms our solution is correct.

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