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Chapter 3: Problem 41

Solve the equation and check your solution. $$4 x-2=3 x+1$$

Short Answer

Expert verified
The solution of the equation \(4x - 2 = 3x + 1\) is \(x = 3\)

Step by step solution

01

Rearrange the equation

Firstly, the aim is to bring the like terms together, which would mean bringing \(3x\) to the left side of the equation and \(2\) to the right side. This is done by subtraction, so subtract \(3x\) from both sides and also subtract \(2\) from both sides to get \(4x - 3x = 1 + 2\)
02

Simplify the equation

After performing the subtraction in Step 1, simplify the resulting equation. This gives us \(x = 3\)
03

Check the solution

Now substitute \(x = 3\) back into the original equation to verify if it satisfies the equation. Substituting in the equation results in \(4(3) - 2 = 3(3) + 1\), which simplifies to \(10 = 10\), a true statement. Thus the solution \(x = 3\) is valid.

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Key Concepts

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

Linear Equations
In mathematics, a linear equation is an equation that forms a straight line when graphed. It typically takes the form \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants.
Linear equations often contain one variable, and the objective is to find the value of that variable that makes the equation true.
To solve a linear equation, you generally perform operations to isolate the variable on one side of the equation.
  • These operations can include addition, subtraction, multiplication, or division.
  • They must be applied to both sides of the equation to maintain balance.
Understanding linear equations is crucial because they appear in many real-world situations.
Whether you're calculating potential earnings, expenses, or even scientific measurements, linear equations help model these relationships.
Checking Solutions
After solving an equation, the next vital step is checking the solution.
This process involves substituting the solution back into the original equation to see if it satisfies the equation.
For instance, in the equation \(4x - 2 = 3x + 1\), after finding \(x = 3\), you substitute \(x\) with 3 in the equation.
  • This involves calculating \(4(3) - 2\) and \(3(3) + 1\).
  • If both sides of the equation equal after substitution, the solution is correct.
Checking solutions is an integral part of solving equations as it confirms the accuracy of your results.
It helps prevent mistakes and ensures that the solution truly satisfies the original equation requirements.
Algebraic Manipulation
Algebraic manipulation is a powerful skill used to transform equations for easier solving.
It involves performing valid operations on equations to rearrange terms and isolate variables.
In our example, \(4x - 2 = 3x + 1\), algebraic manipulation involves:
  • Bringing like terms (those containing \(x\)) together by subtracting \(3x\) from both sides.
  • Eliminating constants from one side, such as by adding 2 to both sides.
  • Simplifying the resulting equation, which ultimately leads to \(x = 3\).
Mastering algebraic manipulation allows you to solve more complex equations efficiently.
It requires understanding and applying different arithmetic operations while respecting the equation's balance. This skill is foundational in algebra and essential for solving a wide variety of mathematical problems.

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Most popular questions from this chapter

Solve and graph the inequality. $$6(3-z) \geq 5(3+z)$$

Translate the verbal statement into a linear inequality. \(x\) is greater than 0 and less than or equal to 6 .

Determine which of the numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers. $$\left\\{-6, \frac{7}{4}, 2,1, \sqrt{49},-8, \frac{4}{3}\right\\}$$

Solve and graph the inequality. $$8(t-3)<4(t-3)$$

Match the statement with the property it represents. (a) Addition Property of Inequality (b) Subtraction Property of Inequality (c) Multiplication Property of Inequality (d) Division Property of Inequality \(-8>-9\), so \(-8+4>-9+4\).
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