For a test question, a mathematics teacher wants to find two constants a and b such that the test item "Simplify a(x + 2y) − b(2x − y)" will have an answer of -4x + 12y. What constants a and b should the teacher use?
a(x + 2y) − b(2x − y) = -4x + 12y
ax + 2ay - 2bx + by = -4x + 12y
x(a-2b) + y(2a + b) = -4x + 12y
so a-2b = -4
and 2a + b = 12
first one times 2 ---> 2a - 4b = -8
second one ----> 2a + b = 12
subtract them: -5b = -20
b = 4
back in first: a - 8= - 4
a = 4
check:
4(x+2y) - 4(2x-y)
= 4x + 8y - 8x + 4y
= -4x + 12y , checks!
awesome thanks!
To find the values of constants a and b that would make the expression "Simplify a(x + 2y) − b(2x − y)" equal to "-4x + 12y," we can set up an equation using the distributive property of multiplication over addition and subtraction.
Let's simplify the given expression step by step:
a(x + 2y) − b(2x − y)
= ax + 2ay − 2bx + by (Applying the distributive property)
Now, let's compare this simplified expression with the expression we want to obtain, which is -4x + 12y. We can match the coefficients of x and y to create a system of equations.
Matching the coefficients of x:
ax - 2bx = -4x
Matching the coefficients of y:
2ay + by = 12y
Now, we need to solve this system of equations simultaneously to find the values of a and b.
For the first equation:
ax - 2bx = -4x
Factoring out x:
x(a - 2b) = -4x
Since this equation should hold for all values of x, we can conclude that a - 2b = -4. (Equation 1)
For the second equation:
2ay + by = 12y
Factoring out y:
y(2a + b) = 12y
Again, since this equation should hold for all values of y, we can conclude that 2a + b = 12. (Equation 2)
Now we have a system of equations:
Equation 1: a - 2b = -4
Equation 2: 2a + b = 12
To solve this system, we can use any method such as substitution or elimination.
Let's use the elimination method. Multiply Equation 1 by 2 and Equation 2 by -1 to eliminate variable a.
2(a - 2b) = -8
-1(2a + b) = -12
Simplifying:
2a - 4b = -8
-2a - b = -12
Now, add the two equations together:
(2a - 4b) + (-2a - b) = -8 + (-12)
Simplifying:
-5b = -20
Divide both sides of the equation by -5:
b = -20 / -5
b = 4
Now that we have the value of b, we can substitute it back into either Equation 1 or Equation 2 to find the value of a.
Using Equation 1:
a - 2b = -4
a - 2(4) = -4
a - 8 = -4
a = -4 + 8
a = 4
Therefore, the constants a and b that the mathematics teacher should use are a = 4 and b = 4 to obtain the desired expression of -4x + 12y.