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Chapter 2: Problem 61

Solve each equation, and check the solution. $$4 x+3 x=21$$

Short Answer

Expert verified
x = 3

Step by step solution

01

Combine Like Terms

Combine the terms with the variable x on the left side of the equation. The equation is: 4x + 3x = 21.
02

Simplify the Equation

Add the coefficients of x together: 4x + 3x = 7x Thus, the equation simplifies to: 7x = 21.
03

Isolate the Variable

To solve for x, divide both sides of the equation by 7: \[ \frac{7x}{7} = \frac{21}{7} \] Simplify to get: x = 3.
04

Check the Solution

Substitute x = 3 back into the original equation to verify: 4(3) + 3(3) = 12 + 9 = 21. Since both sides of the equation are equal, x = 3 is the correct solution.

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Key Concepts

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

Combining Like Terms
When solving a linear equation, the first step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the given equation, we have two terms involving the variable x: 4x and 3x. Since they both contain the same variable, we can add their coefficients together. This step simplifies the equation, making it easier to solve. The equation 4x + 3x becomes 7x when the like terms are combined. So, the initial equation 4x + 3x = 21 simplifies to 7x = 21.
Simplifying the Equation
Simplifying the equation makes it easier to isolate and solve for the variable. After combining like terms, the next step is to perform any necessary arithmetic operations to further simplify the equation. In our case, after combining 4x and 3x to get 7x, the equation becomes 7x = 21. This equation is now much simpler because we can see that solving for x involves straightforward arithmetic.
Isolating the Variable
To find the value of the variable, we need to isolate it on one side of the equation. This typically involves performing inverse operations. For the equation 7x = 21, we want x to be by itself on the left side. To isolate x, we divide both sides of the equation by the coefficient of x, which is 7. So, we get \[ \frac{7x}{7} = \frac{21}{7} \]. This simplifies to x = 3. Now, we have isolated the variable and found its value.
Checking the Solution
Always check your solution by substituting the value back into the original equation. This ensures that your solution is correct. For our equation, we substitute x = 3 back into 4x + 3x = 21. So, we calculate 4(3) + 3(3) which equals 12 + 9, and this equals 21. Since both sides of the equation are equal, our solution x = 3 is verified to be correct. This final step confirms that the answer is accurate and the original equation holds true.

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