In an equilateral triangle, if the lengths of the three sides are 5x − 2, 6x − 10, and 4x + c, what are the values of x and c?

To find the values of x and c, we need to use the fact that all sides of an equilateral triangle are equal in length.

Since the given triangle is equilateral, we can set up the following equation to represent the equality of the three sides:

5x - 2 = 6x - 10 = 4x + c

Now, let's solve this equation step by step.

First, let's equate the first two sides:

5x - 2 = 6x - 10

To solve this equation, we need to isolate the variable x. We can do this by subtracting 5x from both sides:

-2 = x - 10

Next, let's isolate the x-term by adding 10 to both sides:

8 = x

Now that we have the value of x, we can substitute it back into the equation to find the value of c:

4x + c = 4(8) + c

32 + c = 4x + c

Since the three sides are equal, we can equate the second and third sides:

6x - 10 = 4x + c

Substituting the value of x we found earlier:

6(8) - 10 = 4(8) + c

48 - 10 = 32 + c

38 = 32 + c

To isolate the variable c, we can subtract 32 from both sides:

38 - 32 = c

6 = c

Therefore, the value of x is 8, and the value of c is 6.

To find the values of x and c, we can set up and solve an equation based on the properties of an equilateral triangle.

In an equilateral triangle, all three sides have equal lengths. Therefore, we can set up the following equation:

5x - 2 = 6x - 10 = 4x + c

We can solve this equation by equating pairs of the sides and then equating those pairs to the third side.

1) Equating the first and second sides:
5x - 2 = 6x - 10

To solve this equation, we can subtract 5x from both sides:

-2 = x - 10

Next, we can add 10 to both sides:

8 = x

So, x = 8.

2) Equating the first and third sides:
5x - 2 = 4x + c

To solve this equation, we can subtract 4x from both sides:

5x - 4x - 2 = c

Simplifying the left side gives us:

x - 2 = c

Since we found earlier that x = 8, we can substitute this value in:

8 - 2 = c

6 = c

So, c = 6.

Therefore, the values of x and c are 8 and 6, respectively.