At what value of x do the graphs of the equations 2x + y = 6 and 5x + 10y = 10 intersect?

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To find the value of x at which the graphs of the equations 2x + y = 6 and 5x + 10y = 10 intersect, we need to solve the system of equations.

We have:

  • Equation 1: 2x + y = 6
  • Equation 2: 5x + 10y = 10

Let’s first express y from the first equation:

From Equation 1:

y = 6 - 2x

Now, substitute this expression for y into the second equation:

Substituting into Equation 2:

5x + 10(6 - 2x) = 10

Now, simplify this equation:

5x + 60 - 20x = 10
-15x + 60 = 10
-15x = 10 - 60
-15x = -50
x = rac{-50}{-15} = rac{10}{3}

So, the value of x at which the graphs of the two equations intersect is x = rac{10}{3}, or approximately 3.33.

This means that at x = rac{10}{3}, the corresponding value of y can also be found by substituting x back into one of the original equations. Let’s substitute it back into Equation 1 to find y:

y = 6 - 2(rac{10}{3}) = 6 - rac{20}{3} = rac{18}{3} - rac{20}{3} = -rac{2}{3}

Thus, the two equations intersect at the point ( rac{10}{3}, - rac{2}{3}).

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