Evaluate: { \frac{x+5}{4}=\frac{x-3}{3}$}$ (4 Marks)5. A Father Is 4 Times As Old As His Son. After 5 Years, The Father Will Be 3 Times As Old As The Son.
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Introduction
In this problem, we are given an equation involving a variable x. The equation is a rational equation, where the variable x is in the numerator and denominator of fractions. Our goal is to evaluate the given equation and find the value of x.
Step 1: Multiply Both Sides by the Least Common Multiple (LCM)
To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 4 and 3 is 12.
\frac{x+5}{4}=\frac{x-3}{3}
Step 2: Multiply Both Sides by 12
Now, we multiply both sides of the equation by 12 to eliminate the fractions.
12 \times \frac{x+5}{4} = 12 \times \frac{x-3}{3}
Step 3: Simplify the Equation
After multiplying both sides by 12, we can simplify the equation by canceling out the common factors.
3(x+5) = 4(x-3)
Step 4: Expand and Simplify
Next, we expand and simplify the equation by distributing the numbers outside the parentheses to the terms inside.
3x + 15 = 4x - 12
Step 5: Isolate the Variable
Now, we isolate the variable x by moving all the terms containing x to one side of the equation and the constant terms to the other side.
3x - 4x = -12 - 15
Step 6: Simplify and Solve
Finally, we simplify the equation by combining like terms and solving for x.
-x = -27
x = 27
The final answer is .
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Introduction
In this problem, we are given a relationship between the ages of a father and his son. The father is currently 4 times as old as his son. After 5 years, the father will be 3 times as old as the son. Our goal is to find the current age of the son.
Step 1: Let's Represent the Current Ages of the Father and Son
Let's represent the current age of the son as x. Since the father is 4 times as old as his son, the current age of the father is 4x.
Step 2: After 5 Years, the Father's Age Will Be 3 Times the Son's Age
After 5 years, the father's age will be 4x + 5, and the son's age will be x + 5. According to the problem, after 5 years, the father will be 3 times as old as the son. We can write this as an equation:
4x + 5 = 3(x + 5)
Step 3: Expand and Simplify the Equation
Next, we expand and simplify the equation by distributing the numbers outside the parentheses to the terms inside.
4x + 5 = 3x + 15
Step 4: Isolate the Variable
Now, we isolate the variable x by moving all the terms containing x to one side of the equation and the constant terms to the other side.
4x - 3x = 15 - 5
Step 5: Simplify and Solve
Finally, we simplify the equation by combining like terms and solving for x.
x = 10
The final answer is .
Conclusion
In this problem, we used algebraic techniques to solve for the current age of the son. We represented the current ages of the father and son as variables, set up an equation based on the given information, and solved for the variable x. The final answer is 10, which represents the current age of the son.
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Introduction
In this article, we will answer some frequently asked questions about evaluating equations and solving age problems. These questions cover various topics, including algebraic techniques, equation manipulation, and problem-solving strategies.
Q: What is the first step in evaluating an equation?
A: The first step in evaluating an equation is to identify the type of equation and determine the best approach to solve it. This may involve simplifying the equation, isolating the variable, or using algebraic techniques such as factoring or substitution.
Q: How do I simplify an equation?
A: To simplify an equation, you can use various techniques such as combining like terms, canceling out common factors, or using the distributive property. For example, if you have the equation 2x + 5 = 3x - 2, you can simplify it by combining like terms: 2x - 3x = -2 - 5, which simplifies to -x = -7.
Q: What is the least common multiple (LCM) and how do I find it?
A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. To find the LCM, you can list the multiples of each number and find the smallest multiple that appears in both lists. For example, the LCM of 4 and 3 is 12, since 12 is the smallest multiple that appears in both lists.
Q: How do I isolate the variable in an equation?
A: To isolate the variable in an equation, you can use various techniques such as adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides of the equation by the same value. For example, if you have the equation 2x + 5 = 3x - 2, you can isolate the variable x by subtracting 2x from both sides: 5 = x - 2, and then adding 2 to both sides: 7 = x.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation 2x + 5 = 3x - 2 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I solve a system of equations?
A: To solve a system of equations, you can use various techniques such as substitution, elimination, or graphing. For example, if you have two equations: x + 2y = 4 and 2x - 3y = 5, you can solve the system by substitution or elimination.
Q: What is the difference between a direct and an indirect age problem?
A: A direct age problem is a problem in which the age of one person is given, and the age of another person is to be found. For example, if a father is 4 times as old as his son, and the son is 10 years old, then the father is 40 years old. An indirect age problem, on the other hand, is a problem in which the age of one person is not given, but the age of another person is to be found. For example, if a father is 3 times as old as his son after 5 years, and the son is currently 10 years old, then the father is currently 30 years old.
Q: How do I solve an indirect age problem?
A: To solve an indirect age problem, you can use various techniques such as algebraic manipulation, substitution, or elimination. For example, if a father is 3 times as old as his son after 5 years, and the son is currently 10 years old, then you can set up an equation to represent the situation: 3(x + 5) = 4x, where x is the current age of the son. Solving for x, you get x = 10, which is the current age of the son.
Conclusion
In this article, we have answered some frequently asked questions about evaluating equations and solving age problems. We have covered various topics, including algebraic techniques, equation manipulation, and problem-solving strategies. By following these techniques and strategies, you can solve a wide range of equations and age problems.