How do I solve this story problem using either substitution or elimination?
Two friends went on separated hot-air balloon rides. Max Powers was in a hot-air balloon sponsored by Energizer batteries and Rock Bottom was cruising in a hot-air ballon sponsored by Kwik-Krete Concrete. Max took off from the ground rising at a rate of 4 feet per second. At the exact time Max lifted off, Rock was at an altitude 756 feet and descending fast at a rate of 3 feet per second. Some time during the balloon rides, the two friends will be at the same altitude.
1.Write an equation for each balloon that models the balloons height, (h), in terms of time, (t). (Hint: instead of y=mx+b use h=mt+b). Include the real life meaning of the slope and y-intercept of each equation.
2. Solve this system of equations by substitution or elimination. Describe in detail the process of solving this system. Also describe the real-life meaning of your solution.
Bonus: If you were to graph this system of equations, describe where the solution is located.
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1. the slope m for each balloon is the rate of change of altitude. for max, m=4 ft/sec.; for Rock, m=-3 ft/sec. the y-intercept is the value of h at time t=0, or the start of Max's rise and Rock's simultaneous descent from 756 feet. So at t=0, h=0 for Max and h=756 for Rock. Which means, for Max, b=0; for Rock b=756. So the equations are:
2. Solving: since h is already shown as two equivalent values, set them equal to each other:
Add 3t to each side: 7t=756
Divide each side by 7: t=108 (seconds)
Max: h=4(108)=432 feet
Rock: h=-3(108)+756=756-324=432 feet
What this means in reality is that 108 seconds after Max leaves the ground, his balloon will be 432 feet above the ground, and at the same time, Rock's balloon will hav edescended from 756 feet down to 432 feet. At 108 seconds, the two balloons will be at the same atltitude.
Bonus: Graphing this on the h/t coordinate plane would have the lines h=4t and h=-3t+756 intersecting at the point h,t=(432,108).
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