To answer this question, we can use a system of equations. Let's let the number of chairs refinished using method 1 be represented by the variable x, and let the number of chairs refinished using method 2 be represented by the variable y. We know that method 1 (x) takes 1 hour and method 2 (y) takes 2.5 hours. If we use these values as coefficients for their respective variables and set their sum equal to the the total number of hours, we get our first equation:
1x + 2.5y = 157
To get our second equation, we use the same method, except this time using 6 as the coefficient for x, 4 as the coefficient for y, and 568 as the sum of these two terms, as modeled below:
6x + 4y = 568
To solve this system of equations, we can use substitution. To do this, we should solve the first equation for the variable x (get x alone on the left side of the equation):
1x + 2.5y = 157
We should subtract 2.5y from both sides of this equation.
x = 157 - 2.5y
Now, we can substitute this value for x in terms of y into our second equation for the variable x so that we have an equation with only the variable y.
6x + 4y = 568
6(157-2.5y) + 4y = 568
To solve this equation, we must first use the distributive property to get rid of the parentheses on the left side of the equation.
942 - 15y + 4y = 568
Next, we should add the two terms with the variable y together on the left side of the equation.
942 - 11y = 568
After that, we should subtract 942 from both sides of the equation to separate the variables and constants.
-11y = -374
Finally, we should divide both sides of the equation by -11.
y = 34
Now, we can substitute in this value of y into one of our original equations and solve for x.
6x + 4y = 568
6x + 4(34) = 568
Now, we can solve this equation by multiplying together and then subtracting the constants.
6x + 136 = 568
6x = 432
x = 72
Therefore, because x = 72 and y = 34, the furniture shop will refinish 72 chairs using method 1 and 34 chairs using method 2.
Hope this helps!