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Chapter 4: Problem 34

$$ -3 x+2=p(x-q) $$ In the equation above, \(p\) and \(q\) are constants. If there are infinitely many solutions to the equation, what is the value of \(q\) ?

Short Answer

Expert verified
The value of \(q\) such that there are infinitely many solutions to the given equation is \(-\frac{2}{3}\).

Step by step solution

01

Rewrite the equation in standard form

To rewrite the equation in standard form, we want to get all terms on one side of the equation and set the equation to zero. So, we will expand the right side, then move all terms to the left side of the equation:\[ -3x + 2 = p(x - q) \implies -3x + 2 = px - pq \] Next, shift all terms to the left side of the equation:\[ -3x + 2 - px + pq =0 \]
02

Identify conditions for infinitely many solutions

For the equation to have infinitely many solutions, the coefficients of x terms must be equal, and the constant terms must also be equal. This implies the following conditions:\[ -3 = p \] and \[ 2 = pq \]
03

Solve for q

Now we can use these two conditions to find the value of q. Since \(-3 = p\), we can substitute this value into the equation for the constant terms:\[ 2 = (-3)q \] Now, divide both sides by \(-3\) to find the value of \(q\):\[ q = \frac{2}{-3} \] So, the value of \(q\) such that there are infinitely many solutions to the given equation is \(-\frac{2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Solutions
When it comes to equations, having infinite solutions means that there are countless values that satisfy the equation. This typically occurs when the equation represents a line that overlaps with itself, rather than being a single point on the graph.
To have infinite solutions, certain conditions must be met:
  • The equation must simplify to an identity, like \(0 = 0\) or \(c = c\) where \(c\) represents a constant.
  • The coefficients of similar terms should be equal on both sides of the equation.
  • The constant terms must also match on both sides.
In the given exercise, by equating the terms on both sides correctly, we find that satisfying these conditions means adjusting the values of constants, such as \(p\) and \(q\), to ensure an overlap that leads to an infinite number of solutions.
Linear Equations
Linear equations are a fundamental concept in algebra. These are equations of the first degree, meaning they involve only linear terms, which are terms without exponents or radicals. A typical linear equation in one variable has the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
Characteristics of Linear Equations:
  • They graph as straight lines.
  • They have at most one solution unless they are identical (resulting in infinite solutions) or parallel (resulting in no solution).
  • Solving these usually involves isolation of the variable involved.
Linear equations form the basis for many more advanced mathematical concepts. Understanding them helps simplify more complex algebraic tasks. In context with our exercise, the equation \(-3x + 2 = p(x - q)\) is manipulated by moving terms to identify the nature of solutions and constants involved.
Solving for Variables
Solving for variables is one of the main goals when working with equations. It involves isolating the unknown variable on one side of the equation to determine its possible values. This process helps in finding the solution or solutions to the equation.Steps to Solve for Variables:
  • First, simplify the equation if needed by expanding any parentheses or combining like terms.
  • Next, move all terms involving the variable to one side and constant terms to the other side.
  • Finally, isolate the variable by performing operations such as addition, subtraction, multiplication, or division.
In the original solution, solving for the variable \(q\) involved equating the coefficients \(p\) and \(-3\), then solving the resulting simple equation \(2 = (-3)q\). This separation and elimination of terms make it possible to solve for \(q\) efficiently, showing its value when the equation is meant to have infinitely many solutions.

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