Solve the equation and check the solution. Express number.
[tex]$13-3 m+4 m=-4(-2 m-6)+10$[/tex]
The solution set is $\square$ .
Answer (2)
The solution to the equation is m = − 3 . After verifying by substituting back into the original equation, both sides confirm that the solution is correct. Therefore, the solution set is { − 3 } .
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Simplify the given equation by combining like terms: 13 + m = − 4 ( − 2 m − 6 ) + 10 .
Distribute and simplify further: 13 + m = 8 m + 24 + 10 , which becomes 13 + m = 8 m + 34 .
Isolate m by subtracting m and 34 from both sides: − 21 = 7 m .
Solve for m by dividing both sides by 7 : m = − 3 . The solution set is { − 3 } .
Explanation
Problem Setup We are given the equation 13 − 3 m + 4 m = − 4 ( − 2 m − 6 ) + 10 and we want to solve for m .
Simplifying Left Side First, we simplify both sides of the equation. On the left side, we combine the terms with m : − 3 m + 4 m = m . So the left side becomes 13 + m .
Simplifying Right Side On the right side, we distribute the − 4 to the terms inside the parentheses: − 4 ( − 2 m − 6 ) = 8 m + 24 . So the right side becomes 8 m + 24 + 10 .
Combining Constants Now we combine the constant terms on the right side: 24 + 10 = 34 . So the right side simplifies to 8 m + 34 .
Isolating m Now our equation is 13 + m = 8 m + 34 . We want to isolate m on one side of the equation. Let's subtract m from both sides: 13 = 7 m + 34 .
Further Isolation Next, we subtract 34 from both sides: 13 − 34 = 7 m , which simplifies to − 21 = 7 m .
Solving for m Finally, we divide both sides by 7 to solve for m : m = 7 − 21 = − 3 .
Checking the Solution So, m = − 3 . Now we need to check our solution by substituting m = − 3 back into the original equation.
Left Side Evaluation Left side: 13 − 3 ( − 3 ) + 4 ( − 3 ) = 13 + 9 − 12 = 22 − 12 = 10 .
Right Side Evaluation Right side: − 4 ( − 2 ( − 3 ) − 6 ) + 10 = − 4 ( 6 − 6 ) + 10 = − 4 ( 0 ) + 10 = 0 + 10 = 10 .
Final Answer Since the left side equals the right side ( 10 = 10 ), our solution m = − 3 is correct. Therefore, the solution set is { − 3 } .
Examples
In physics, this type of equation could represent a scenario where you're trying to find the equilibrium position of an object acted upon by multiple forces. Each term in the equation could represent a force, and solving for 'm' would give you the position where the forces balance out, resulting in no net force and thus equilibrium. Understanding how to solve such equations is crucial in many engineering and scientific applications.
