Problem 19 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$4 s-4=30-s ; 7$$

Step-by-Step Solution

Answer

No, 7 is not a solution to the equation.

Step 1: Substitute the provided number for 's'

Substitute 7 for 's' in the equation. So, the equation \(4s - 4 = 30 - s\) becomes \(4*7 - 4 = 30 - 7\).

Step 2: Solve each side of the Equation

Calculate the left and right side of the equation separately. On the left, \(4*7 - 4\) equals 24. On the right, \(30 - 7\) equals 23.

Step 3: Compare the Results

Now, compare the mathematical results from both sides of the equation. Here, the left side (24) does not equal the right side (23).

Key Concepts

Check Solutions

When you check solutions for equations, you aim to verify whether a given number truly satisfies the equation. This means inserting the number into the equation in place of the variable and checking if both sides of the equation become equal.
In our example, we are given the equation \(4s - 4 = 30 - s\) and asked to check if \(s = 7\) is a solution. By substituting 7 for \(s\), we can form the equation \(4 \times 7 - 4 = 30 - 7\).
After performing the calculations, if both sides of the equation produce the same result, the number is indeed a solution. However, if they result in different values, as seen with 24 and 23 in this case, \( s = 7 \) is not a solution.
Checking solutions helps confirm the accuracy of equation-solving and ensures that you understand how to work with the variables. Always make sure to compute carefully and compare both sides accurately.

Substitution Method

The substitution method is a fundamental technique used when solving equations, especially useful for verifying potential solutions to an equation. It involves replacing a variable with a given number, allowing us to determine if this number satisfies the equation entirely.
In the context of the given problem, substituting 7 for \(s\) in the equation \(4s - 4 = 30 - s\) helped us to check if 7 makes the equation true.
By calculating both sides separately with this specific value, we convert the problem from an abstract algebraic form into a simple arithmetic task. This substitution made it clear that 24 is not equal to 23, thus proving that \(s = 7\) does not satisfy the equation.

• This method is not just useful for checking specific solutions but is also the first step in solving more complex systems of equations.
• When correctly applied, the substitution method can eliminate variables systematically.

Substitution is crucial to build solid foundational skills in equation manipulation.

Linear Equations

Linear equations, such as \(4s - 4 = 30 - s\), are algebraic equations where each term is either a constant or the product of a constant and a single variable. They form the basis for understanding more complex algebraic structures.
These equations are termed "linear" because, if graphed, they form a straight line, showing the direct relationship between variables. In a typical form, linear equations can be written as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
Solving linear equations involves finding the value of the variable that makes the equation true, which means making both sides equal. Understanding linear equations is critical as it teaches you how to solve for unknowns and apply mathematical operations systematically.

• Linear equations serve as a stepping stone to more advanced topics like quadratic equations and linear inequalities.
• They are essential in various real-world applications, from calculating budgets to predicting trends in data analysis.

Mastery of linear equations aids in building higher-level problem-solving skills.