Problem 32
Question
Check to see if x = 5 is or is not a solution of the equation or the inequality. $$ 3 x+2 x=25 $$
Step-by-Step Solution
Verified Answer
Yes, x = 5 is a solution to the equation \(3x + 2x = 25\).
1Step 1: Simplify The Equation
Combine like terms on the left side of the equation. This simplifies to \(5x = 25\).
2Step 2: Substitute The Value
Substitute the given value of x in the simplified equation. That makes it \(5*5 = 25\), which simplifies to \(25 = 25\).
3Step 3: Verify The Solution
Verify that both sides of the equation are equal. Since 25 equals 25, x=5 is a solution to the equation \(3x + 2x = 25\).
Key Concepts
Equation SimplificationSubstitution MethodVerifying Solutions
Equation Simplification
Understanding how to simplify an algebraic equation is a fundamental skill in solving mathematical problems. When working with equations, the goal is often to isolate the variable we are trying to solve for. Before we can get to that point, we must simplify the equation by combining like terms. Like terms refer to terms in the equation that have the exact same variable raised to the same power. For example, in the equation \(3x + 2x = 25\), both terms on the left side of the equation have the variable \(x\) and can thus be combined.
By adding \(3x\) and \(2x\) together, we simplify the equation to \(5x = 25\). This is an essential step because it streamlines the equation, making the following steps more manageable. Simplification does not change the solution to the equation; it merely prepares it for the next phase—substitution or further manipulation to isolate the variable.
By adding \(3x\) and \(2x\) together, we simplify the equation to \(5x = 25\). This is an essential step because it streamlines the equation, making the following steps more manageable. Simplification does not change the solution to the equation; it merely prepares it for the next phase—substitution or further manipulation to isolate the variable.
Substitution Method
Once an equation is simplified, the substitution method can often be employed to find the solution to the equation. This technique involves replacing the variable with a number to see if the equation makes a true statement.
For instance, if we substitute \(x\) with 5 in the equation \(5x = 25\), we get \(5 \times 5\), which equals 25. In this case, when we substitute 5 for \(x\), the equation holds true, suggesting that 5 is indeed the solution. Substitution is not only useful for solving equations but also for verifying that a proposed solution is correct. In contrast, if the substitution results in an equation that does not equate, it shows that the substituted value is not a solution. It's essential to carefully perform the substitution step to avoid any errors that might lead you to incorrect conclusions.
For instance, if we substitute \(x\) with 5 in the equation \(5x = 25\), we get \(5 \times 5\), which equals 25. In this case, when we substitute 5 for \(x\), the equation holds true, suggesting that 5 is indeed the solution. Substitution is not only useful for solving equations but also for verifying that a proposed solution is correct. In contrast, if the substitution results in an equation that does not equate, it shows that the substituted value is not a solution. It's essential to carefully perform the substitution step to avoid any errors that might lead you to incorrect conclusions.
Verifying Solutions
The last step in solving an algebraic equation is to verify that the obtained solution satisfies the original equation. This act of verification confirms the accuracy of our solution. In our example exercise, after determining that \(x = 5\) is a potential solution, we verify by substituting 5 back into the original equation \(3x + 2x = 25\).
After calculation, we observe that the left-hand side of the equation equates to the right-hand side, fulfilling the condition for a valid solution: \(5 \times 5 = 25\) simplifies indeed to \(25 = 25\). A true verification means that both sides match perfectly, thus validating the solution. It's crucial to perform this final check to ensure the solution is correct, as any mistake in the earlier steps could lead to an incorrect 'solution' being accepted without challenge.
After calculation, we observe that the left-hand side of the equation equates to the right-hand side, fulfilling the condition for a valid solution: \(5 \times 5 = 25\) simplifies indeed to \(25 = 25\). A true verification means that both sides match perfectly, thus validating the solution. It's crucial to perform this final check to ensure the solution is correct, as any mistake in the earlier steps could lead to an incorrect 'solution' being accepted without challenge.
Other exercises in this chapter
Problem 32
Evaluate the variable expression when a = 3 and c = 5. $$ a^{2}+c^{2} $$
View solution Problem 32
Check to see if the given value of the variable is or is not a solution of the equation or the inequality. $$16+x^{2} \div 4=17 ; x=2$$
View solution Problem 32
Evaluate the expression. \(2 \cdot 3^{2}-7\)
View solution Problem 32
Evaluate the expression for then given value of the variable. \(h^{5}\) when \(h=2\)
View solution