Solve $5(x-4)=2(x+5)$ for $x$
Answer (2)
Expand both sides of the equation: 5 x − 20 = 2 x + 10 .
Subtract 2 x from both sides: 3 x − 20 = 10 .
Add 20 to both sides: 3 x = 30 .
Divide both sides by 3 : x = 10 . The solution is 10 .
Explanation
Understanding the Problem We are given the equation 5 ( x − 4 ) = 2 ( x + 5 ) and we need to solve for x . This is a linear equation, and we will use algebraic manipulation to isolate x on one side of the equation.
Expanding Both Sides First, we expand both sides of the equation by distributing the constants: 5 ( x − 4 ) = 5 x − 20 2 ( x + 5 ) = 2 x + 10 So the equation becomes: 5 x − 20 = 2 x + 10
Isolating x Terms Next, we want to isolate the x terms on one side of the equation. We can subtract 2 x from both sides: 5 x − 2 x − 20 = 2 x − 2 x + 10 3 x − 20 = 10
Isolating Constant Terms Now, we want to isolate the constant terms on the other side of the equation. We can add 20 to both sides: 3 x − 20 + 20 = 10 + 20 3 x = 30
Solving for x Finally, we can solve for x by dividing both sides by 3 :
3 3 x = 3 30 x = 10
Verification To verify the solution, we substitute x = 10 back into the original equation: 5 ( 10 − 4 ) = 2 ( 10 + 5 ) 5 ( 6 ) = 2 ( 15 ) 30 = 30 Since the equation holds true, our solution is correct.
Final Answer Therefore, the solution to the equation 5 ( x − 4 ) = 2 ( x + 5 ) is x = 10 .
Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items, determining the distance traveled at a constant speed, or converting between different units of measurement. For example, if you want to determine how many hours you need to work to earn a certain amount of money, you can set up a linear equation to model the situation and solve for the number of hours. Understanding how to solve linear equations is a fundamental skill in mathematics and has practical applications in everyday life.
The solution to the equation 5 ( x − 4 ) = 2 ( x + 5 ) is x = 10 . This is found by expanding, isolating the variable, and solving through simple algebraic steps. Verification confirms that the solution is correct.
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