Which is a reasonable first step that can be used to solve the equation $4 x+3(x+2)=5(2 x-3)$?
A. Distribute the 3 to $x+2$, and distribute the 5 to $(2 x-3)$.
B. Combine $4 x$ and $x$ on the left side of the equation.
C. Apply the subtraction property of equality to move the 2 from the left to the right side of the equation.
D. Apply the addition property of equality to move the 3 from the right to the left side of the equation.
Answer (2)
The most reasonable first step to solve the equation 4 x + 3 ( x + 2 ) = 5 ( 2 x − 3 ) is to distribute the 3 to x + 2 and the 5 to 2 x − 3 . This simplifies the equation to 4 x + 3 x + 6 = 10 x − 15 . Therefore, the selected option is A.
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Distribute the 3 to x + 2 , resulting in 3 x + 6 .
Distribute the 5 to ( 2 x − 3 ) , resulting in 10 x − 15 .
Simplify the equation to 4 x + 3 x + 6 = 10 x − 15 .
The reasonable first step is to distribute the 3 and the 5. Distribute the 3 to x + 2 , and distribute the 5 to ( 2 x − 3 ) .
Explanation
Analyze the Problem The problem presents an equation, 4 x + 3 ( x + 2 ) = 5 ( 2 x − 3 ) , and asks for a reasonable first step to solve it. The given options include distributing, combining like terms, and applying properties of equality.
Strategy When solving equations, a good first step is often to simplify each side of the equation as much as possible before attempting to isolate the variable. This typically involves expanding any terms within parentheses using the distributive property.
Apply the Distributive Property The first option, 'Distribute the 3 to x + 2 , and distribute the 5 to ( 2 x − 3 ) ', aligns with this strategy. Distributing the 3 into ( x + 2 ) gives 3 x + 6 , and distributing the 5 into ( 2 x − 3 ) gives 10 x − 15 . This simplifies the equation to 4 x + 3 x + 6 = 10 x − 15 , which is easier to work with.
Evaluate Other Options The other options are not as reasonable as a first step. Combining 4 x and x is premature since the distributive property hasn't been applied yet. Applying the subtraction or addition property of equality to move constants around is also best done after simplifying each side.
Conclusion Therefore, the most reasonable first step is to distribute the 3 to ( x + 2 ) and distribute the 5 to ( 2 x − 3 ) .
Examples
In real-world scenarios, this type of equation solving is used in various fields such as physics, engineering, and economics. For instance, when calculating the trajectory of a projectile, you might encounter an equation that needs simplification before you can solve for the unknown variable, such as time or distance. Similarly, in economics, you might need to solve equations to determine the equilibrium price in a market, which often involves simplifying expressions first. Let's say you are trying to determine how many items you need to sell to break even. You have fixed costs of $500 and each item costs $5 to produce. You sell each item for $10. The equation would be 10 x = 5 x + 500 . The first step would be to simplify the equation by subtracting 5 x from both sides to get 5 x = 500 .
