Problem 11
Question
Find all values of \(x\) satisfying the given conditions. \(y_{1}=13 x-4, y_{2}=5 x+10,\) and \(y_{1}\) exceeds \(y_{2}\) by 2.
Step-by-Step Solution
Verified Answer
The only value of \(x\) that satisfies these conditions is \(x=2\)
1Step 1: Formulate the equations based on the given condition
We know from the problem that \(y_{1}\) exceeds \(y_{2}\) by 2. Hence, we can write it as \(y_{1}=y_{2}+2\). Substituting the given \(y_{1}\) and \(y_{2}\) values into this equation, we get \(13x-4=5x+10+2\)
2Step 2: Simplify the equation
The equation can be simplified by combining like terms on both sides. It simplifies to \(13x-5x=4+10+2\) which further simplifies to \(8x=16\)
3Step 3: Solve for the variable
To solve for \(x\), divide both sides of the equation by 8. This results in \(x=16/8\)
4Step 4: Final answer
The solution for \(x\) resulting from the division operation is the final answer, \(x=2\)
Key Concepts
Algebraic ManipulationSystems of EquationsSubstitution Method
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying mathematical expressions and equations to solve for unknown variables. This can involve a variety of operations, including addition, subtraction, multiplication, division, factoring, expanding, and combining like terms.
In the exercise provided, algebraic manipulation comes into play when the equations are simplified. After setting up the equation based on the given condition, the equation is manipulated by subtracting the term with the variable on both sides, and afterward, combining the constant terms. This simplification is a fundamental step and it taps into the heart of algebra – the ability to transform an equation into a simpler form without changing its solution.
In the exercise provided, algebraic manipulation comes into play when the equations are simplified. After setting up the equation based on the given condition, the equation is manipulated by subtracting the term with the variable on both sides, and afterward, combining the constant terms. This simplification is a fundamental step and it taps into the heart of algebra – the ability to transform an equation into a simpler form without changing its solution.
- The expression \(13x - 4 = 5x + 12\) is simplified by subtracting \(5x\) from both sides to get \(8x - 4 = 12\).
- Then, adding the constants together and moving them to one side of the equation provides a clearer picture of what \(x\) must be.
Systems of Equations
A system of equations is a set of two or more equations with the same set of variables. In the context of the problem, we are dealing with a very simple system where we have two expressions for \(y_1\) and \(y_2\) and an additional condition relating them.
Understanding how to work with systems of equations is crucial for solving many real-world problems. The key objective is to find a solution that satisfies all equations simultaneously. In our example, the system implied by the problem is:
Although we are not dealing with a system where we have to solve for two variables, the premise is the same. The last condition provides the necessary relation to allow one equation to be used to solve for the single variable, \(x\).
Understanding how to work with systems of equations is crucial for solving many real-world problems. The key objective is to find a solution that satisfies all equations simultaneously. In our example, the system implied by the problem is:
- \[ y_1 = 13x - 4 \]
- \[ y_2 = 5x + 10 \]
- \[ y_1 = y_2 + 2 \]
Although we are not dealing with a system where we have to solve for two variables, the premise is the same. The last condition provides the necessary relation to allow one equation to be used to solve for the single variable, \(x\).
Substitution Method
The substitution method is one way to solve systems of equations. This method involves solving one equation for one variable and then substituting the resulting expression into the other equation(s).
In our case, the process begins by recognizing that we have values for \(y_1\) and \(y_2\), along with a relationship between them. We express \(y_1\) in terms of \(y_2\) using the given condition \(y_1 = y_2 + 2\). We then substitute the expressions for \(y_1\) and \(y_2\) from the conditions given in the exercise into this equation. This provides us with an equation in one variable, which we then manipulate algebraically to solve for \(x\).
In our case, the process begins by recognizing that we have values for \(y_1\) and \(y_2\), along with a relationship between them. We express \(y_1\) in terms of \(y_2\) using the given condition \(y_1 = y_2 + 2\). We then substitute the expressions for \(y_1\) and \(y_2\) from the conditions given in the exercise into this equation. This provides us with an equation in one variable, which we then manipulate algebraically to solve for \(x\).
- Once we substitute, we get: \(13x-4 = 5x + 10 + 2\).
- Now, we've effectively reduced the system to a single variable equation, which simplifies the solution process.
Other exercises in this chapter
Problem 11
In Exercises \(9-20,\) find each product and write the result in standard form. $$(-5+4 i)(3+i)$$
View solution Problem 11
Solve each equation in Exercises \(1-14\) by factoring. $$ 2 x(x-3)=5 x^{2}-7 x $$
View solution Problem 11
Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 3)$$
View solution Problem 11
Check all proposed solutions. \(\sqrt{3 x+18}=x\)
View solution