Solve: $\frac{2(x-3)}{5}=\frac{4(x+2)}{7}$
Answer (1)
Multiply both sides by 5 and 7 to get rid of the fractions: 14 ( x − 3 ) = 20 ( x + 2 ) .
Expand both sides: 14 x − 42 = 20 x + 40 .
Rearrange the equation to isolate x: − 6 x = 82 .
Solve for x: x = − 3 41 .
Explanation
Problem Analysis We are given the equation 5 2 ( x − 3 ) = 7 4 ( x + 2 ) and we want to solve for x .
Eliminating Fractions First, we multiply both sides of the equation by 5 and 7 to eliminate the fractions. This gives us: 7 ⋅ 2 ( x − 3 ) = 5 ⋅ 4 ( x + 2 ) Simplifying, we get: 14 ( x − 3 ) = 20 ( x + 2 )
Expanding the Equation Next, we expand both sides of the equation by distributing the constants: 14 x − 42 = 20 x + 40
Isolating x Terms Now, we want to isolate the terms with x on one side and the constant terms on the other side. Subtracting 14 x from both sides gives: − 42 = 6 x + 40 Subtracting 40 from both sides gives: − 82 = 6 x
Solving for x Finally, we divide both sides by 6 to solve for x : x = 6 − 82 Simplifying the fraction by dividing both the numerator and the denominator by 2, we get: x = − 3 41
Final Answer Therefore, the solution to the equation is x = − 3 41 .
Examples
Solving linear equations is a fundamental skill in algebra and is used in various real-life situations. For example, if you want to determine how many hours you need to work to earn a certain amount of money, or if you need to calculate the amount of ingredients needed to scale a recipe, you would use linear equations. In this case, solving the equation helped us find the value of x that satisfies the given condition. Understanding how to manipulate and solve equations is crucial for problem-solving in many areas of life.
