Problem 58
Question
Find all values of \(x\) such that \(y=0\) $$y=2[3 x-(4 x-6)]-5(x-6)$$
Step-by-Step Solution
Verified Answer
The value of \(x\) that makes \(y=0\) is \(x=6\)
1Step 1: Simplify the Equation
Start by simplifying the equation. First distribute the innermost parentheses: \(y=2[3x-4x+6]-5x+30\). Then distribute the 2 across the brackets: \(y=2*-x+12-5x+30\). Combine like terms to get a standard polynomial equation: \(y=-7x+42\).
2Step 2: Set the Equation Equal to Zero
Since we want to find the roots of the equation, set \(y\) equal to zero and solve for \(x\): \(0=-7x+42\).
3Step 3: Solve for x
To solve for \(x\), first add \(7x\) to both sides of equation to get \(7x=42\). Then divide both sides of the equation by 7 to isolate \(x\). The solution is \(x=6\).
Key Concepts
Polynomial Root FindingAlgebraic Expressions SimplificationEquation Solving Step by Step
Polynomial Root Finding
The process of polynomial root finding is an essential algebraic skill that involves identifying the value(s) of the variable that make a polynomial equation equal to zero. These values are known as the roots or solutions of the polynomial.
The task often starts with setting the polynomial equal to zero and solving for the variable of interest. For a polynomial in the form of ax^n + bx^(n-1) + ... + k = 0, where a, b, ..., k are constants and n is a positive integer, the goal is to find all possible values of x that satisfy the equation. These values where the graph of the polynomial touches or crosses the x-axis are of particular interest in various fields such as mathematics, physics, and engineering.
In our exercise, the polynomial equation given is y = 2[3x - (4x - 6)] - 5(x - 6). To find the root, we need to simplify the polynomial and then set it equal to zero to solve for x.
The task often starts with setting the polynomial equal to zero and solving for the variable of interest. For a polynomial in the form of ax^n + bx^(n-1) + ... + k = 0, where a, b, ..., k are constants and n is a positive integer, the goal is to find all possible values of x that satisfy the equation. These values where the graph of the polynomial touches or crosses the x-axis are of particular interest in various fields such as mathematics, physics, and engineering.
In our exercise, the polynomial equation given is y = 2[3x - (4x - 6)] - 5(x - 6). To find the root, we need to simplify the polynomial and then set it equal to zero to solve for x.
Algebraic Expressions Simplification
Simplification of algebraic expressions is a fundamental process in algebra which involves reducing expressions to their simplest form. This is accomplished by performing operations like distributing multiplicative factors across parentheses, combining like terms, and reducing fractions where possible. Simplification can make equations easier to understand and work with, and is a critical step before solving.
For example, in the given exercise, the expression 2[3x-(4x-6)]-5(x-6) looks complex at first glance. To simplify, we distribute the 2 into the bracket and add/subtract like terms, eventually arriving at y = -7x + 42, a much simpler form. The importance of this step lies in its ability to transform the problem into a clear, linear equation which can be solved with basic algebraic techniques.
For example, in the given exercise, the expression 2[3x-(4x-6)]-5(x-6) looks complex at first glance. To simplify, we distribute the 2 into the bracket and add/subtract like terms, eventually arriving at y = -7x + 42, a much simpler form. The importance of this step lies in its ability to transform the problem into a clear, linear equation which can be solved with basic algebraic techniques.
Equation Solving Step by Step
The step by step approach to equation solving provides a systematic way to find the solutions to algebraic equations. This methodical process ensures that no detail is overlooked and that complicated problems can be tackled in manageable chunks.
The steps typically begin with simplifying the algebraic expression, as seen in the first section. Following this, the next steps might involve rearranging the equation to isolate the variable (e.g., getting the x's on one side and the constants on the other), performing arithmetic operations such as addition or subtraction to both sides to maintain equality, and finally, multiplying or dividing to solve for the variable.
In our example, once we simplified to y = -7x + 42, we set y to zero and solve the resulting equation 0 = -7x + 42. By adding 7x to both sides and then dividing by 7, we find that x = 6. This stepwise progression from more complex to more simple makes the task of solving equations more accessible and less intimidating.
The steps typically begin with simplifying the algebraic expression, as seen in the first section. Following this, the next steps might involve rearranging the equation to isolate the variable (e.g., getting the x's on one side and the constants on the other), performing arithmetic operations such as addition or subtraction to both sides to maintain equality, and finally, multiplying or dividing to solve for the variable.
In our example, once we simplified to y = -7x + 42, we set y to zero and solve the resulting equation 0 = -7x + 42. By adding 7x to both sides and then dividing by 7, we find that x = 6. This stepwise progression from more complex to more simple makes the task of solving equations more accessible and less intimidating.
Other exercises in this chapter
Problem 58
Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(x^{2}-2 x\right)^{2}-11\left(x^{2}-2 x\right)+24=0$$
View solution Problem 58
Write an original word problem that can be solved using a linear equation. Then solve the problem.
View solution Problem 59
In Exercises 59–94, solve each absolute value inequality. $$ |x|
View solution Problem 59
Solve equation by completing the square. $$ 2 x^{2}-7 x+3=0 $$
View solution