\[ 2x + 3 = -\frac{1}{2}x + 1 \]

["# How to Solve the Equation ( 2x + 3 = -\frac{1}{2}x + 1 ): A Step-by-Step Guide", "Solving linear equations is a fundamental skill in algebra, essential for mastering more complex mathematical concepts. One common equation students encounter is:", "[\n2x + 3 = -\frac{1}{2}x + 1\n]", "Understanding how to solve this equation not only helps in homework and exams but also strengthens problem-solving skills applicable in many real-world scenarios. This article provides a clear, step-by-step solution to help you confidently tackle equations of this type.", "## Step 1: Eliminate Fractions to Simplify\nThe equation contains a fraction—(-\frac{1}{2}x)—which can be made easier by eliminating the fraction. Multiply every term in the equation by 2:", "[\n2 \cdot (2x + 3) = 2 \cdot \left(-\frac{1}{2}x + 1\right)\n]", "This gives:\n[\n4x + 6 = -x + 2\n]", "Now the equation is simpler to work with, with no fractions remaining.", "## Step 2: Collect Like Terms\nOur goal is to gather all variable terms ((x)) on one side and constant terms on the other. Start by adding (x) to both sides to eliminate (-x):", "[\n4x + x + 6 = 2\n]", "[\n5x + 6 = 2\n]", "Next, subtract 6 from both sides:", "[\n5x = 2 - 6\n]", "[\n5x = -4\n]", "## Step 3: Solve for (x)\nTo isolate (x), divide both sides by 5:", "[\nx = \frac{-4}{5}\n]", "Thus, the solution to the equation is:", "[\nx = -\frac{4}{5}\n]", "## Step 4: Verify the Solution\nIt’s always good practice to check your answer by substituting (x = -\frac{4}{5}) back into the original equation:", "Left side:\n[\n2\left(-\frac{4}{5}\right) + 3 = -\frac{8}{5} + 3 = -\frac{8}{5} + \frac{15}{5} = \frac{7}{5}\n]", "Right side:\n[\n-\frac{1}{2}\left(-\frac{4}{5}\right) + 1 = \frac{4}{10} + 1 = \frac{2}{5} + 1 = \frac{2}{5} + \frac{5}{5} = \frac{7}{5}\n]", "Both sides equal (\frac{7}{5}), confirming that the solution is correct.", "## Why This Equation Matters\nThe equation (2x + 3 = -\frac{1}{2}x + 1) is a classic example of a linear equation with rational coefficients. Solving it reinforces key algebraic techniques such as:", "- Eliminating fractions\n- Collecting like terms\n- Isolating variables\n- Verifying solutions", "These skills are essential for higher-level math, including systems of equations, graphing linear functions, and modeling real-life problems in science, finance, and engineering.", "## Practice Problem\nTry solving the equation below for yourself:\n[\n2x + 3 = -\frac{1}{2}x + \frac{5}{2}\n]\nFollow similar steps: eliminate the fraction by multiplying by 2, collect like terms, and isolate (x).", "---", "Summary:\nSolving (2x + 3 = -\frac{1}{2}x + 1) involves clearing fractions, combining terms, isolating the variable, and verifying the solution. Mastering this process helps build a strong foundation in algebra and prepares you for more advanced mathematical challenges. Keep practicing to gain confidence!"]