/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 \(3 x+7=2 x+4\)... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 53

\(3 x+7=2 x+4\)

Short Answer

Expert verified
The solution is \(x = -3\).

Step by step solution

01

Subtract \(2x\) from both sides

Start by isolating the term containing \(x\) on one side of the equation. Subtract \(2x\) from both sides: \(3x + 7 - 2x = 2x + 4 - 2x\). This simplifies to: \(x + 7 = 4\).
02

Subtract \(7\) from both sides

Now, isolate \(x\) by subtracting \(7\) from both sides: \(x + 7 - 7 = 4 - 7\). This simplifies to: \(x = -3\).
03

Verify the solution

To ensure the solution is accurate, substitute \(x = -3\) back into the original equation: \(3(-3) + 7 = 2(-3) + 4\). Simplifying both sides results in \(-9 + 7 = -6 + 4\), which further simplifies to \(-2 = -2\). The solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolation of Variables
When solving linear equations, the primary goal is to isolate the variable (often represented as x). This means getting the variable on one side of the equation and everything else on the other side.
To start, look at the given equation: \[3x + 7 = 2x + 4\]We'll move all terms involving x to one side. Subtract 2x from both sides:\[3x + 7 - 2x = 2x + 4 - 2x\]This gives:\[x + 7 = 4\]Now, we need to isolate x further. To do this, subtract 7 from both sides:\[x + 7 - 7 = 4 - 7\]This leaves us with:\[x = -3\]
Verify Solutions
Once we have isolated the variable and found a solution, it is essential to verify that the solution is correct. Verification ensures that no mistakes were made.
To verify x = -3, we substitute it back into the original equation:\[3(-3) + 7 = 2(-3) + 4\]Simplify each side step by step:\[-9 + 7 = -6 + 4\]Further simplify:\[-2 = -2\]Since both sides of the equation are equal, the solution x = -3 is verified and correct.
Simplifying Equations
Simplifying equations is a critical step that makes solving them much easier. Simplification involves combining like terms and performing arithmetic operations to reduce the equation to a simpler form.
In the original equation:\[3x + 7 = 2x + 4\]we simplified by subtracting 2x from both sides:\[3x + 7 - 2x = 2x + 4 - 2x\]This resulted in:\[x + 7 = 4\]Next, we simplified further by subtracting 7 from both sides:\[x + 7 - 7 = 4 - 7\]Leading to the simplest form:\[x = -3\]Each step makes the equation easier to solve, breaking down the problem into manageable parts.

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Most popular questions from this chapter

To see the connection between the solution of an equation and the solutions of the corresponding inequalities. Solve the following inequality, and graph the solution set on a number line. \( 3 x+2>14 \)

Solve each formula for the specified variable. \(A x+B y=C\) for \(y\)

Solve each equation for \(y\) $$ -3 x+5 y=-15 $$

Find the measure of an angle whose complement is five times its measure.

If 4 is added to twice a number and this sum is multiplied by \(2,\) the result is the same as if the number is multiplied by 3 and 4 is added to the product. What is the number?
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