Solve the equation for the given variable. If your answer is a fraction, write it in reduced, fractional form. Do NOT convert the answer to a decimal. [tex]$4(3 y+1)=-38$[/tex]
Answer (2)
Distribute the 4: 4 ( 3 y + 1 ) = 12 y + 4 .
Subtract 4 from both sides: 12 y = − 42 .
Divide by 12: y = 12 − 42 .
Simplify the fraction: y = − 2 7 .
Explanation
Problem Analysis We are given the equation 4 ( 3 y + 1 ) = − 38 and we need to solve for y . The goal is to isolate y on one side of the equation.
Distribute First, distribute the 4 on the left side of the equation:
4 ( 3 y + 1 ) = 4 × 3 y + 4 × 1 = 12 y + 4
So the equation becomes:
12 y + 4 = − 38
Isolate the y term Next, subtract 4 from both sides of the equation to isolate the term with y :
12 y + 4 − 4 = − 38 − 4
12 y = − 42
Solve for y Now, divide both sides by 12 to solve for y :
y = 12 − 42
Simplify the fraction Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 42 and 12 is 6.
y = 12 ÷ 6 − 42 ÷ 6 = 2 − 7
So, y = − 2 7
Final Answer Therefore, the solution to the equation is y = − 2 7 .
Examples
Imagine you're baking a cake and need to adjust a recipe that serves 4 people to serve a different number. Solving equations like this helps you scale ingredients accurately. For instance, if the original recipe uses 4 tablespoons of sugar and you want to use -38 tablespoons, you can set up an equation to find out how much of each ingredient you need to adjust. This kind of proportional reasoning is essential for cooking, mixing chemicals, or any situation where you need to scale quantities.
To solve the equation 4 ( 3 y + 1 ) = − 38 , we distribute the 4, subtract 4 from both sides, divide by 12, and simplify the fraction to find y = − 2 7 .
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