Solve and check the linear equation. [tex]4(x-2)+15=3(x+4)[/tex] What is the solution? Select the correct choice below and, if necessary, fill in the answer. A. The solution set is {$\square$} B. The solution set is {[tex]x | x[/tex] is a real number}. C. The solution set is $\varnothing$.
Answer (2)
Expand both sides of the equation: 4 ( x − 2 ) + 15 = 4 x − 8 + 15 and 3 ( x + 4 ) = 3 x + 12 .
Simplify both sides: 4 x + 7 = 3 x + 12 .
Isolate x : x = 5 .
The solution set is 5 .
Explanation
Analyze the problem We are given the equation 4 ( x − 2 ) + 15 = 3 ( x + 4 ) and we need to solve for x . This is a linear equation, so we will use algebraic manipulation to isolate x on one side of the equation.
Expand both sides First, we expand both sides of the equation using the distributive property: 4 ( x − 2 ) + 15 = 4 x − 8 + 15 3 ( x + 4 ) = 3 x + 12 So the equation becomes: 4 x − 8 + 15 = 3 x + 12
Simplify both sides Next, we simplify both sides by combining like terms: 4 x + 7 = 3 x + 12
Isolate x Now, we want to isolate x on one side of the equation. We can subtract 3 x from both sides: 4 x − 3 x + 7 = 3 x − 3 x + 12 x + 7 = 12
Solve for x Finally, we subtract 7 from both sides to solve for x :
x + 7 − 7 = 12 − 7 x = 5
Check the solution To check our solution, we substitute x = 5 back into the original equation: 4 ( 5 − 2 ) + 15 = 3 ( 5 + 4 ) 4 ( 3 ) + 15 = 3 ( 9 ) 12 + 15 = 27 27 = 27 Since the equation holds true, our solution is correct.
State the solution The solution to the equation is x = 5 . Therefore, the solution set is {5}.
Examples
Linear equations are used in everyday life to calculate things like the cost of items, distances, and time. For example, if you know that a taxi charges a flat fee of $5 plus 2 p er mi l e , yo u c an u se a l in e a re q u a t i o n t o d e t er min e t h e t o t a l cos t o f a r i d e . I f t h er i d e i s 10 mi l es , t h ee q u a t i o n w o u l d b e y = 2x + 5 , w h ere x i s t h e n u mb ero f mi l es an d y i s t h e t o t a l cos t . Pl ugg in g in 10 f or x , w e g e t y = 2(10) + 5 = 25$. So the total cost of the 10-mile taxi ride is $25.
The solution to the linear equation 4 ( x − 2 ) + 15 = 3 ( x + 4 ) is x = 5 , so the solution set is { 5 }. We confirmed this by substituting back into the original equation. Thus, the correct choice is A: The solution set is { 5 }.
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