Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Jon and Valeria each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Jon has 28 signatures, and Valeria has 10. Jon is collecting signatures at an average rate of 6 per day, whereas Valeria is averaging 15 signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have? In _____ days, Jon and Valeria will each have collected _____ signatures.

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.  Jon and Valeria each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Jon has 28 signatures, and Valeria has 10. Jon is collecting signatures at an average rate of 6 per day, whereas Valeria is averaging 15 signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have?   In _____ days, Jon and Valeria will each have collected _____ signatures.

Bytelearn · Accepted Answer

Set Up Equations: Let x represent the number of days and y represent the number of signatures collected. We can set up two equations based on the given rates of signature collection for Jon and Valeria.  For Jon: Starting signatures: 28 Rate of collection: 6 signatures per day Equation: y = 6x + 28  For Valeria: Starting signatures: 10 Rate of collection: 15 signatures per day Equation: y = 15x + 10  Now we have a system of equations: 1. y = 6x + 28 2. y = 15x + 10Set Equations Equal: To find the point where Jon and Valeria have the same number of signatures, we set the two equations equal to each other:  6x + 28 = 15x + 10Solve for x: Now we solve for x by subtracting 6x from both sides of the equation:  6x + 28 - 6x = 15x + 10 - 6x 28 = 9x + 10Isolate x: Next, we subtract 10 from both sides to isolate the term with x:  28 - 10 = 9x + 10 - 10 18 = 9xFind Value of x: To find the value of x, we divide both sides by 9:  18 / 9 = 9x / 9 2 = x  So, it will take 2 days for Jon and Valeria to have the same number of signatures.Substitute x into Equation: Now we need to find the number of signatures y that they will each have after 2 days. We can substitute x = 2 into either of the original equations. We'll use Jon's equation:  y = 6x + 28 y = 6(2) + 28 y = 12 + 28 y = 40  So, after 2 days, Jon and Valeria will each have collected 40 signatures.

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