Substitute into the first equation: \(2x + 3(4x - 5) = 6\).

["# Substitute into the First Equation: Solving (2x + 3(4x - 5) = 6) with Confidence", "Complex algebraic equations often involve substitution techniques to simplify expressions and solve for variables efficiently. One common step students must master is substituting parts of an equation to reduce complexity—especially when dealing with parentheses and linear terms. In this article, we explore how to substitute into the first equation of the classic step-by-step process:\n[2x + 3(4x - 5) = 6]", "By strategically substituting internal expressions, we streamline the equation and solve it with clarity and speed. Let’s break down this substitution method and apply it to solve the equation confidently.", "---", "## Step-by-Step Breakdown of the Equation", "We begin with:\n[\n2x + 3(4x - 5) = 6\n]", "### What Does Substitution Mean Here?", "Substitution in algebra means replacing a complex or repeated expression with a simpler symbol or value to simplify manipulation. Here, the expression ( (4x - 5) ) appears inside parentheses and is multiplied by 3. Instead of expanding fully at once, we substitute this entire inner expression to keep working with a manageable form.", "---", "## Step 1: Identify the Substitution Point", "Notice the parentheses:\n[\n3(4x - 5)\n]", "This entire term can be substituted temporarily with a variable—say ( y )—to simplify our work:\nLet\n[\ny = 4x - 5\n]", "Then the original equation becomes:\n[\n2x + 3y = 6\n]", "Now the equation contains only simpler components: (2x), (3y), and the constant (6).", "---", "## Step 2: Express One Variable in Terms of the Other", "Although strict substitution uses replacement without rewriting, understanding what substitution enables is key. From ( y = 4x - 5 ), we solve for (x) to substitute back, but for now, keep it as ( y ) in our transformed equation.", "We still have (x) in (2x), so express (x) using (y):", "[\ny = 4x - 5 \Rightarrow 4x = y + 5 \Rightarrow x = \frac{y + 5}{4}\n]", "Substitute this into (2x):\n[\n2x = 2 \cdot \frac{y + 5}{4} = \frac{y + 5}{2}\n]", "Now substitute into the full equation:\n[\n\frac{y + 5}{2} + 3y = 6\n]", "---", "## Step 3: Solve the Simplified Equation", "Multiply every term by 2 to eliminate the denominator:\n[\n2 \cdot \frac{y + 5}{2} + 2 \cdot 3y = 2 \cdot 6\n\Rightarrow (y + 5) + 6y = 12\n]", "Simplify:\n[\ny + 5 + 6y = 12 \Rightarrow 7y + 5 = 12\n]", "Subtract 5 from both sides:\n[\n7y = 7\n]", "Divide by 7:\n[\ny = 1\n]", "---", "## Step 4: Back-Substitute to Find (x)", "Recall ( y = 4x - 5 ) and ( y = 1 ), so:\n[\n4x - 5 = 1 \Rightarrow 4x = 6 \Rightarrow x = \frac{6}{4} = \frac{3}{2}\n]", "---", "## Why Substitute in This Way?", "Substituting ( y = 4x - 5 ) allows us to treat the parenthetical expression as a single variable, simplifying arithmetic and minimizing errors. While formal substitution replaces parts of the equation, this method preserves clarity and makes combining like terms smoother—especially helpful in more complex equations.", "---", "## Final Answer", "The solution to the equation\n[\n2x + 3(4x - 5) = 6\n]\nis\n[\nx = \frac{3}{2}\n]", "---", "## SEO Keywords to Boost Your Article Rank:", "- Substitute into algebraic equations\n- Solve linear equations step-by-step\n- Algebra substitution examples\n- Step-by-step solving first degree equation\n- Simplify equations with parentheses\n- Solve ( 2x + 3(4x - 5) = 6 <br/>\n- Improve algebra problem-solving skills", "---", "## Conclusion", "Substituting into the first equation transforms complexity into simplicity, enabling efficient solving. Whether you’re a student mastering algebra or a teacher explaining methods—knowing how to substitute within expressions like ( (4x - 5) ) builds strong problem-solving habits. Practice substitution daily—it’s a powerful tool that unlocks advanced algebra techniques."]