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# 4.1: Simultaneous Equations

Difficulty Level: At Grade Created by: CK-12

Name: __________________

## Distance, Rate, and Time

1. You set off walking from your house at 2miles\begin{align*}2\;\mathrm{miles}\end{align*} per hour.

a. Fill in the following table.

After this much time (t)You have gone this far (d)12 hour1 hour2 hours3 hours10 hours\begin{align*}& \text{After this much time}\ (t) && \frac{1}{2}\ \text{hour} && 1 \ \text{hour} && 2\ \text{hours} && 3\ \text{hours} && 10\ \text{hours}\\ & \text{You have gone this far}\ (d)\\\end{align*}

b. Write the function d(t)\begin{align*}d(t)\end{align*}.

2. You set off driving from your house at 60miles\begin{align*}60\;\mathrm{miles}\end{align*} per hour.

a. Fill in the following table.

After this much time (t)You have gone this far (d)12 hour1 hour2 hours3 hours10 hours\begin{align*}& \text{After this much time}\ (t) && \frac{1}{2}\ \text{hour} && 1 \ \text{hour} && 2\ \text{hours} && 3\ \text{hours} && 10\ \text{hours}\\ & \text{You have gone this far}\ (d)\\\end{align*}

b. Write the function d(t)\begin{align*}d(t)\end{align*}.

3. You set off in a rocket, flying upward at 200miles\begin{align*}200\;\mathrm{miles}\end{align*} per hour.

a. Fill in the following table.

After this much time (t)You have gone this far (d)12 hour1 hour2 hours3 hours10 hours\begin{align*}& \text{After this much time}\ (t) && \frac{1}{2}\ \text{hour} && 1 \ \text{hour} && 2\ \text{hours} && 3\ \text{hours} && 10\ \text{hours}\\ & \text{You have gone this far}\ (d)\\\end{align*} b. Write the function d(t)\begin{align*}d(t)\end{align*}.

4. Write the general relationship between distance traveled (d)\begin{align*}(d)\end{align*}, rate (r)\begin{align*}(r)\end{align*}, and time (t)\begin{align*}(t)\end{align*}.

5. You start off for school at 55mph\begin{align*}55\;\mathrm{mph}\end{align*}. 15\begin{align*}\frac{1}{5}\end{align*} of an hour later, your mother realizes you forgot your lunch. She dashes off after you, at 70mph\begin{align*}70\;\mathrm{mph}\end{align*}. Somewhere on the road, she catches up with you, throws your lunch from her car into yours, and vanishes out of sight.

Let d\begin{align*}d\end{align*} equal the distance from your home where your mother catches up with you. Let t\begin{align*}t\end{align*} equal the time that you took to reach that distance. (Note that you and your mother traveled the same distance, but in different time.) d\begin{align*}d\end{align*} should be measured in miles, and t\begin{align*}t\end{align*} in hours (not minutes).

a. Write the distance-rate-time relationship for you, from the time you leave the house until your mother catches up with you.

b. Write the distance-rate-time relationship for your mother, from the time she leaves the house until she catches up with you.

c. Based on those two equations, can you figure out how far you were from the house when your mother caught you?

Name: __________________

Homework: Simultaneous Equations by Graphing

In each problem, find all x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} values that satisfy both conditions. Your answers may be approximate.

1. 2y=6x+10\begin{align*}2y=6x+10\end{align*} and 3y=12x+9\begin{align*}3y=12x+9\end{align*}

a. Put both equations into y=mx+b\begin{align*}y=mx+b\end{align*} format. Then graph them.

b. List all points of intersection.

c. Check these points to make sure they satisfy both equations.

2. y=2x3\begin{align*}y=2x-3\end{align*} and 3y=6x+3\begin{align*}3y=6x+3\end{align*}

a. Put both equations into y=mx+b\begin{align*}y=mx+b\end{align*} format. Then graph them.

b. List all points of intersection.

c. Check these points to make sure they satisfy both equations.

3. \begin{align*}y=x-3\end{align*} and \begin{align*}2y=2x-6\end{align*}

a. Put both equations into \begin{align*}y=mx+b\end{align*} format. Then graph them on the back.

b. List all points of intersection.

c. Check these points to make sure they satisfy both equations.

4. \begin{align*}y=x\end{align*} and \begin{align*}y=x^2-1\end{align*}

a. Graph them both on the back.

b. List all points of intersection.

c. Check these points to make sure they satisfy both equations.

5. \begin{align*}y=x^2+2\end{align*} and \begin{align*}y=x\end{align*}

a. Graph them both on the back.

b. List all points of intersection.

c. Check these points to make sure they satisfy both equations.

6. \begin{align*}y=x^2+4\end{align*} and \begin{align*}y=2x+3\end{align*}

a. Put the second equation into \begin{align*}y=mx+b\end{align*} format. Then graph them both on the back.

b. List all points of intersection.

c. Check these points to make sure they satisfy both equations.

7. Time for some generalizations...

a. When graphing two lines, is it possible for them to never meet? _______ To meet exactly once? _________ To meet exactly twice? _________ To meet more than twice? ___________

b. When graphing a line and a parabola, is it possible for them to never meet? _______ To meet exactly once? _________ To meet exactly twice? _________ To meet more than twice? ___________

This last problem does not involve two lines, or a line and a parabola: it’s a bit weirder than that. It is the only problem on this sheet that should require a calculator.

8. \begin{align*} y =\frac{6x}{x^2+1}\end{align*} and \begin{align*}y = 4\sqrt{x} - 5\end{align*}

a. Graph them both on your calculator and find the point of intersection as accurately as you can.

b. Check this point to make sure it satisfies both equations.

Name: __________________

Simultaneous Equations

1. Emily is hosting a major after-school party. The principal has imposed two restrictions. First (because of the fire codes) the total number of people attending (teachers and students combined) must be \begin{align*}56\end{align*}. Second (for obvious reasons) there must be one teacher for every seven students. How many students and how many teachers are invited to the party?
1. Name and clearly identify the variables.
2. Write the equations that relate these variables.
3. Solve. Your final answers should be complete English sentences (not “the answer is \begin{align*}2\end{align*},” but “there were \begin{align*}2\end{align*} students there.” Except it won’t be \begin{align*}2\end{align*}. You get the idea, right?)
2. A group of \begin{align*}75\end{align*} civic-minded students and teachers are out in the field, picking sweet potatoes for the needy. Working in the field, Kasey picks three times as many sweet potatoes as Davis—and then, on the way back to the car, she picks up five more sweet potatoes than that! Looking at her newly increased pile, Davis remarks “Wow, you’ve got \begin{align*}29\end{align*} more potatoes than I do!” How many sweet potatoes did Kasey and Davis each pick?
1. Name and clearly identify the variables.
2. Write the equations that relate these variables.
3. A hundred ants are marching into an anthill at a slow, even pace of \begin{align*}2\;\mathrm{miles}\end{align*} per hour. Every ant is carrying either one bread crumb, or two pieces of grass. There are \begin{align*}28\end{align*} more bread crumbs than there are pieces of grass. How many of each are there?
1. Name and clearly identify the variables.
2. Write the equations that relate these variables.
4. Donald is \begin{align*}14\;\mathrm{years}\end{align*} older than Alice. \begin{align*}22\;\mathrm{years}\end{align*} ago, she was only half as old as he was. How old are they today?
1. Name and clearly identify the variables.
2. Write the equations that relate these variables.
5. Make up your own word problem like the ones above, and solve it.
6. \begin{align*}3x - 2y & = 16\\ 7x - y & = 30\end{align*}
1. Solve by substitution
2. Solve by elimination
7. \begin{align*}3x + 2y & = 26\\ 2x + 4y & = 32\end{align*}
1. Solve by substitution
2. Solve by elimination
8. Under what circumstances is substitution easiest?
9. Under what circumstances is elimination easiest?

Name: __________________

Homework: Simultaneous Equations

1. \begin{align*}6x + 2y & = 6\\ x - y & = 5\end{align*}
1. Solve by substitution.
2. \begin{align*}3x + 2y & = 26\\ 2x - 4y & = 4\end{align*}
1. Solve by elimination.
3. \begin{align*}2y - x & = 4\\ 2x + 20y & = 4\end{align*}
1. Solve any way you like.
4. \begin{align*}3x + 4y & = 12\\ 5x - 3y & = 20\end{align*}
1. Solve any way you like.
5. \begin{align*}ax + y & = 6\\ 2x + y & = 4\end{align*}
1. Solve any way you like. You are solving for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}; \begin{align*}a\end{align*} is just a constant. (So your final answer will say \begin{align*}x=\mathrm{blah-blah}\end{align*}, \begin{align*}y=\mathrm{blah-blah}\end{align*}.” The blah-blah will both have a in them.)

The “Generic” Simultaneous Equations

Here is the generic simultaneous equations.

\begin{align*}ax + by & = e\\ cx + dy & = f\end{align*}

I call them “generic” because every possible pair of simultaneous equations looks exactly like that, except with numbers instead of \begin{align*}a, b, c, d, e,\end{align*} and \begin{align*}f\end{align*}. We are going to solve these equations.

Very important!!! When I say by “solve it” I mean, find a formula \begin{align*}x=\mathrm{blah-blah}\end{align*} where the blah-blah has only \begin{align*}a, b, c, d, e,\end{align*} and \begin{align*}f\end{align*}: no \begin{align*}x\end{align*} or \begin{align*}y\end{align*}. And, of course, \begin{align*}y=\end{align*} some different formula with only \begin{align*}a, b, c, d, e,\end{align*} and \begin{align*}f\end{align*}. If we can do that, we will be able to use these formulas to immediately solve any pair of simultaneous equations, just by plugging in the numbers.

We can solve this by elimination or by substitution.I am going to solve for \begin{align*}y\end{align*} by elimination. I will use all the exact same steps we have always used in class.

Step 1: Make the coefficients of \begin{align*}x\end{align*} line up

To do this, I will multiply the top equation by \begin{align*}c\end{align*} and the bottom equation by \begin{align*}a\end{align*}.

\begin{align*}acx + bcy & =ec\\ acx + ady & = af\end{align*}

Step 2: Subtract the second equation from the first

This will make the \begin{align*}x\end{align*} terms go away.

\begin{align*}& \quad acx \ + \ bcy \ \ = \ ec\\ & \underline{-(acx \ + \ ady) = af}\\ & \qquad bcy-ady \ = ec-af\end{align*}

Step 3: Solve for \begin{align*}y\end{align*}

This is something we’ve done many times in class, right? First, pull out a \begin{align*}y\end{align*}; then divide by what is in parentheses.

\begin{align*}bcy - ady & = ec - af\\ y(bc - ad) & = ec - af\\ y & = \frac{ec - ad}{bc - ad}\end{align*}

So what did we do?

We have come up with a totally generic formula for finding \begin{align*}y\end{align*} in any simultaneous equations. For instance, suppose we have...

\begin{align*}3x + 4y & = 18\\ 5x + 2y & = 16\end{align*}

We now have a new way of solving this equation: just plug into \begin{align*}y = \frac{ec - af}{bc - ad}\end{align*}. That will tell us that \begin{align*}y = \frac{(18)(5) - (3)(16)}{(4)(5) - (3)(2)} = \frac{90 - 48}{20 - 6} = \frac{42}{14} = 3\end{align*}.

Didja get it?

Here’s how to find out.

1. Do the whole thing again, starting with the generic simultaneous equations, except solve for \begin{align*}x\end{align*} instead of \begin{align*}y\end{align*}.
2. Use your formula to find \begin{align*}x\end{align*} in the two equations I did at the bottom (under “So what did we do?”)
3. Test your answer by plugging your \begin{align*}x\end{align*} and my \begin{align*}y=3\end{align*} into those equations to see if they work!

Name: __________________

Sample Test: 2 Equations and 2 Unknowns

1. Evan digs into his pocket to see how much pizza he can afford.He has \begin{align*}\3.00\end{align*}, exactly enough for two slices. But it is all in dimes and nickels! Counting carefully, Evan discovers that he has twice as many dimes as nickels.

a. Identify and clearly label the variables.

b. Write two equations that represent the two statements in the question.

c. Solve these equations to find how many nickels and dimes Jeremy has.

2. Black Bart and the Sheriff are having a gunfight at high noon. They stand back to back, and start walking away from each other: Bart at \begin{align*}4\;\mathrm{feet}\end{align*} per second, the Sheriff at \begin{align*}6\;\mathrm{feet}\end{align*} per second. When they turn around to shoot, they find that they are \begin{align*}55\;\mathrm{feet}\end{align*} away from each other.

a. Write the equation \begin{align*}d=rt\end{align*} for Bart.

b. Write the equation \begin{align*}d=rt\end{align*} for the Sheriff.

c. Solve, to answer the question: for how long did they walk away from each other?

d. How far did Bart walk?

3. Mrs. Verbatim the English teacher always assigns \begin{align*}5\end{align*} short stories (\begin{align*}2,000\end{align*} words each) for every novel (\begin{align*}60,000\end{align*} words each) that she assigns. This year she has decided to assign a total of \begin{align*}350,000\end{align*} words of reading to her students. How many books and how many short stories should she select?

a. Identify and clearly label the variables.

b. Write two equations that represent the two conditions that Mrs. V imposed.

c. Solve these equations to find the number of works she will be assigning.

4. Solve by graphing. Answers may be approximate. (But use the graph paper and get as close as you can.)

\begin{align*}y &= x^2 - 3\\ y &= -|x| + 2\end{align*}

5. Solve, using substitution.

\begin{align*}3x + y &= - 2\\ 6x - 2y &= 12\end{align*}

6. Solve, using elimination.

\begin{align*}2x + 3y & = -11\\ 3x - 6y &= 4 \frac{1}{2}\end{align*}

7. Solve any way you like.

\begin{align*}2x &= 6y + 12\\ x - 9 &= 3y\end{align*}

8. Solve any way you like.

\begin{align*}2y + 3x &= 20\\ \frac{1}{2}y + x &= 6\end{align*}

9. a. Solve for \begin{align*}x\end{align*}. (*No credit without showing your work!)

\begin{align*}ax + by &= e\\ cx + dy &= f\end{align*}

b. Use the formula you just derived to find \begin{align*}x\end{align*} in these equations.

\begin{align*}3x + 4y &= 7\\ 2x + 3y &= 11\end{align*}

Extra credit:

Redo number \begin{align*}9\end{align*}. If you used elimination before, use substitution. If you used substitution, use elimination.

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