Solve for $x$ in the following equation: $-2 y=\frac{2}{3} x+6 z$. Do not write $x=$ in your answer.
Answer (1)
Subtract 6 z from both sides: − 2 y − 6 z = 3 2 x .
Multiply both sides by 2 3 : 2 3 ( − 2 y − 6 z ) = x .
Simplify: − 3 y − 9 z = x .
The solution is: − 3 y − 9 z .
Explanation
Isolating the x term We are given the equation − 2 y = 3 2 x + 6 z and we want to solve for x . This means we want to isolate x on one side of the equation.
Subtracting 6z First, subtract 6 z from both sides of the equation to isolate the term with x : − 2 y − 6 z = 3 2 x
Multiplying by 3/2 Next, multiply both sides of the equation by 2 3 to solve for x : 2 3 ( − 2 y − 6 z ) = 2 3 ⋅ 3 2 x − 3 y − 9 z = x
Final Answer Therefore, x = − 3 y − 9 z .
Examples
In physics, if you know the relationship between force, mass, and acceleration but need to find the mass given the other two, you would rearrange the formula F = ma to solve for m , resulting in m = a F . Similarly, in economics, if you have a linear cost function C = a Q + b (where C is the total cost, Q is the quantity, a is the variable cost per unit, and b is the fixed cost) and you need to find the quantity Q given the total cost C , you would rearrange the formula to solve for Q , resulting in Q = a C − b . These rearrangements are essential for making predictions and decisions based on available data.
