How do you first set up a linear equation and then how do you solve it?

Question

Kobe Bryant holds the 2nd highest record for the highest points ever scored in one game.His record is made up of 2-point shots and 3-point shots. Kobe total points was 81 points with 36 shots made. How many 2-point shots did he make?

Answer 1

Let's start by breaking up the problem's information. We know there are two different types of shots Kobe can make: a 2 point shot, and a 3 point shot. The total amount of points Kobe made was 81 points and it took 36 shots to get that many points. We don't know what amount of 2 point or 3 point shots were made to add up to 81 points though. This is why setting up a system of linear equations is a good tool to help us figure that out.

Let's make some variables to help us separate what a 2 point shot is and a 3 point shot is:

We'll call the 2 point shot x

We'll call the 3 point shot y

We know that the total of shots made adds up to 36. Be careful not to confuse this with the amount of points earned. You can make a shot but you may not always get points. Let's make this into our first equation:

x + y = 36 (Equation 1)

Now, we also know that the amount of points made was 81. Because the shots that went in get points, we can multiply our 2 point shots (x) by 2, and our 3 point shots (y) by 3. The products of these variables and their constants will have a sum of 81. Let's make that our second equation:

2*x + 3*y = 81 (Equation 2)

How do we know we have enough equations? Simple. Count the number of variables we don't know the value of that exist in our equations. There are only 2: x and y. If we have the amount of equations as the same amount of unknown variables, we can solve for the unknowns. In this case, we have 2 unknowns, and 2 equations, so now we can solve.

One possible way to solve is by isolating one variable first in one equation. We can put one variable in terms of a different variable this way. Then, we can take that isolated variable, plug it into a different equation, and solve for the value of at least one variable. That value can be plugged into another equation to get the other value we need.

Let's take equation to, and make it equal to an equation in terms of x:

x + y = 36

y = (36 - x)

Now we can take this manipulated equation and put it into equation 1 wherever we have a y:

2x + 3y = 81

2x + 3*(36-x) = 81

Be careful about multiplying: make sure to use the distributive property for this.

2x + 108 - 3x = 81

Now we can simplify this equation and solve for the value of x:

108 - x = 81

-x = -27

We have a negative on both sides, so that can be canceled out too.

x = 27

Now we can take our value of x and plug it back into equation 2 to find the value of y:

x + y = 36

27 + y = 36

y = 9

We finally have answers of x = 27 and y = 9. But we're done. We need to recall what we know x as and what we know y as. x represents 2 point shots and y represents 3 point shots. So we need to label our answers as that.

In the end, Kobe made 27 2 point shots and 9 3 point shots.

We can check our answers by plugging our values of x and y into both equations to see if both sides of our equation we use are equal to each other. That's a good tool to have when finishing a system of linear equations problem.

Hope this helps!

Answer 2

x represents the number of 2-point shots

36 - x represents the number of 3-;point shots

2x + 3(36 - x) = 81

Can you solve for x and answer?