Substitute \( x = \frac{3}{2} \) into \( 2x + 3y = 6 \): \( 2 \times \frac{3}{2} + 3y = 6 \).
["# Solving the Equation: Substitute ( x = \frac{3}{2} ) into ( 2x + 3y = 6 )", "When solving linear equations, substitution is a powerful and straightforward method. One common substitution is using a specific value to simplify the equation and find the corresponding variable value. In this article, we’ll explore what happens when we substitute ( x = \frac{3}{2} ) into the equation ( 2x + 3y = 6 ).", "## The Equation We Start With", "The original equation is:\n[\n2x + 3y = 6\n]", "## Substituting ( x = \frac{3}{2} )", "Now, replace ( x ) with ( \frac{3}{2} ):\n[\n2 \left( \frac{3}{2} \right) + 3y = 6\n]", "### Step-by-Step Calculation", "First, calculate ( 2 \ imes \frac{3}{2} ):\n[\n2 \ imes \frac{3}{2} = \frac{6}{2} = 3\n]", "Now substitute back into the equation:\n[\n3 + 3y = 6\n]", "### Solving for ( y )", "Next, isolate ( 3y ) by subtracting 3 from both sides:\n[\n3y = 6 - 3 = 3\n]", "Divide both sides by 3:\n[\ny = \frac{3}{3} = 1\n]", "## The Solution", "By substituting ( x = \frac{3}{2} ) into ( 2x + 3y = 6 ), we find that:\n[\ny = 1\n]", "This confirms that ( \left( \frac{3}{2}, 1 \right) ) is the solution to the equation.", "## Why This Matters for Algebra and Real-World Problems", "Using substitution with specific values helps verify solutions quickly, particularly in systems of equations or applied problems like budgeting and physics. It is a foundational technique that builds confidence in algebraic manipulation and problem-solving.", "## Key Takeaways", "- Substitution simplifies solving for unknown variables.\n- Replacing ( x ) with ( \frac{3}{2} ) streamlined the equation clearly.\n- Solving ( 2 \ imes \frac{3}{2} + 3y = 6 ) gives ( y = 1 ).\n- This approach is essential for mastering linear equations.", "---", "Practice tip: Try solving similar equations by picking other values for ( x ) to deepen your understanding of substitution and linear systems."]
