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Lesson 6: Two-step equation word problems

# Two-step equation word problem: garden

Here's a nifty word problem in which we find the dimensions of a garden given only the perimeter. Let's create an equation to solve! Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• I am having some trouble figuring out how to make an expression out of a word problem. What is the best way for me to find the expression in a problem?
• I try to teach my students to code sentences, that is write letters, operations, equals, and other things above words and see what happens. So verbs are often equal signs, words like more than, greater, bigger, etc. write a +, words like twice (2*), three times (3*), product, etc. show multiply. If you have a more than (+) or less than (-), the expressions flip, so 6 more than twice a number is 2x + 6. With this, an expression or equation often become easy to write.
• how do you know to determine an equation? Because I watched the video at least 5 times and still don't understand
• In this video we are asked for the dimensions of a rectangle when given the perimeter, so the heart of the equation is the…
Perimeter Formula,
P = 2W + 2L
the sum of all the sides
, the measure of the shape's outline. (It is expected we have this formula memorized.)

P = perimeter
W = width
L = length

P = 2W + 2L
=
P = W + W + L + L

★The word problem gives us…
Perimeter = 60
so let's substitute that in…
=
60 = W + W + L + L

★We're given the hint:
the length is twice as long as the width

So…
Length = 2 • Width
=
L = 2W

Now we can replace the L variables in the equation with 2W, which makes the equation solvable for width.

60 = W + W + L + L
substitute each L with 2W
=
60 = W + W + 2W + 2W
combine like terms
=
60 = 6W
The variable is being multiplied by six, use Opposite Operation of division to isolate variable, divide both sides by six
=
60/6 = 6W/6
=
10 = W ←yay! 🥳 the Width

★Now we need the Length, which we already know is two times the Width!
L = 2 • 10
=
L = 20 ←yay! 🥳 the Length

★Dimensions of the rectangle:
Width = 10
Length = 20

We can check and know these are the correct values by plugging them back into the Perimeter Formula.

(ㆁωㆁ) Hope this helps!
• My Question is
When you have a variables and numbers together in the problem how do you know what to do or what do you do first....?
• This is probably a little late but i think i can help. ( I interpreted your question like this) ex.-11b+7=40. Thats an example of an equation that I think you're talking about. So when you are solving this take everything you learned about the order of operations and throw it out the window. The first thing you are going to do is try to isolate the variable. ( This is the way that I was taught) so first you are going to reverse the operation of adding 7. So you will subtract 7 by 7 getting you to 0. And anything you do to one side you have to do to the opposite side so you will subtract 7 by 40 (40-7). So now that you have isolated the variable you should have this. . . . -11b=33. The next thing to do would be to divide -11 by-11. You would get 0 but still have the b. Then you would do the same to the other side. so you would divide (33/-11) and you should get -3. So now you should have b=-3.

Hope this helps!
• both the length and width, in feet, of a rectangular garden are integers. If the perimeter of the garden is 24 feet, what is the greatest possible area, in square feet of the garden
• For a rectangle of a given perimeter, the area is maximized by making the length and width as close together as possible. Since 24 is divisible by 4, we can make the length and width equal even though the problem states that they have to be integers. So we can use a square with side length 24/4 = 6 feet, which then has area 6^2 = 36 square feet. This is the maximum possible area.
• Um... I already know how to SOLVE the equation, what I'm having trouble with is writing the equation itself. Could someone please help?
• You could think about writing the equation as width plus length plus width plus length equal the perimeter. or if you use variables for width and length you could say W + L + W + L = P and a statement in the question tells us that length is equal to 2 times the width so we can substitute for L as the value of 2 times W. this would give us W + 2W + W + 2W = P

Now if we combine like terms (all of the W's are like terms) we end up with 6W = P and the question tells us that P is equal to 60 ft. Once we plug the values in we get 6W = 60. Now we can use the inverse property of multiplication (which is division) to solve for W. We can divide 6 times W by 6; and we also have to do that to the other side so we have 6W/6 = 60/6; then we get the value of W which equals 10 ft.

Now we can substitute what we know about length (L). We know that L is equal to 2W. And we know that 2W is 2 times 10. so the value of L is equal to 2 times 10 which is 20 ft.

Then we check the values of what we know about perimeter. That is W + L + W + L = P. Plug in the values we get 10 + 20 + 10 + 20 = 60; since both sides equal 60 the equation is true.

Hope that helped!
• the area of a rectangular garden is 240 sq ft, and the length of its diagonal is 26ft. find the dimensions of the garden. *draw a picture of the situation, be sure to identify your variables, then set up a system of nonlinear equations and use the substitution method to solve*
• The rectangular garden will be the shape of a rectangle with two unknown sides that you will need to solve. Let's call it length L and width W. The diagonal will divide the rectangle into two right triangles with base L and height W.

To write your equations, it will obviously involve the sides L and W. But what else are given?
The area of the rectangle is 240sq ft. What relations between L and W does this give? This will be your first equation.
The other given info is the diagonal line is 26 and it forms a right triangle. Given a right triangle, the first thing that comes to my mind is the parthagorean theorem, a^2 + b^2 = c^2. Using this, in term of L and W, you can write your second equation.
Then use substitution to solve for L and W, the dimensions of the rectangular garden.
• What is the perimeter.
• perimeter circle the t and underline rim. so you are adding around the rim (outside) of a figure.
• I'm kind of confused, why do we divide both sides with 6? Is it because we need to cancel out the 6w to find out the answer?
• You should not cancel out the 6w, only the 6 as the coefficient of w.
• Does "at least" look like look like a less than or equal to?
• At least would be a minimum, so it would be a x ≥ # which is greater than or equal to.
• Tim answered all the questions on his math test but got 10 answers wrong. He received 4 points for every correct answer, and there was no penalty for wrong answers. His score was 76 points.
Write an equation to determine the total number of questions (q)(q)left parenthesis, q, right parenthesis on Tim's math test.
• Because Tim answered 10 of the q questions wrong and answered all the questions, he answered (q-10) questions right.

Since each right answer was 4 points and there was no penalty for wrong answers, Tim’s score was 4(q-10) points.

Since Tim’s score was 76 points, a correct equation is 4(q-10) = 76.

Have a blessed, wonderful Christmas holiday!

## Video transcript

The perimeter of Tina's rectangular garden is 60 feet. If the length of the garden is twice the width, what are the dimensions of the garden? So let's draw this garden here, Tina's garden. So it's a rectangle. They tell us that it's a rectangular garden. So it looks something like this. And let's say that this is the width. So if this is the width, then this is also going to be the width. And this is the length up here. And they tell us that the length of the garden is twice the width. So if this is w, then the length is going to be 2w. It's going to be twice the width. This is also going to be 2w over here. Now, what's the perimeter of this garden? Well, it's going to be w plus w plus 2w plus 2w. Let me write this down. The perimeter of this garden is going to be equal to w plus 2w plus w plus 2w, which is equal to what? This is w plus 2w is 3w, 4w, 6w. So this is equal to 6w. That's the perimeter in terms of the width. But they also tell us that the actual numerical value of the perimeter is 60 feet. It is 60 feet. So this perimeter 6w must be equal to 60 if we assume that we're dealing with feet. So we just have the equation 6w is equal to 60. We can divide both sides of this equation by 6 so that we have just a w on the left-hand side. 6w divided by 6 is just w. And then 60 divided by 6 is 10. So we have w is equal to 10. So the width of the garden is 10. So this distance over here is 10. And then what is the length of the garden? Well, it's 2 times the width. So this is equal to 20. The length is equal to 20. And so we're done. This is a 20 by 10 garden.