Problem 57
Question
The equations in contain small geometric figures that represent nonzero real numbers. Use the multiplication property of equality to isolate \(x\) on one side of the equation and the geometric figures on the other side. $$\Delta=-x$$
Step-by-Step Solution
Verified Answer
The solution to the equation \(\Delta = -x\) is \(x = -\Delta\).
1Step 1: Understanding the provided equation
The provided equation is \(\Delta = -x\). Here, the goal is to isolate \(x\) on one side of the equation to compute its value.
2Step 2: Applying the multiplication property of equality
The multiplication property of equality allows you to multiply both sides of an equation by the same number, without changing the equality. Now, apply it to the equation. Multiply both sides of the equation by -1 to reverse the sign and isolate \(x\). Doing so, the equation becomes: \(-1*\Delta = -1*(-x)\).
3Step 3: Simplifying the equation
After the multiplication, the equation simplifies to \(-\Delta = x\), or when rephrasing, \(x = -\Delta\). So, \(x\) equals \(-\Delta\), which means that the value of \(x\) is the negative of whatever value \(\Delta\) represents.
Key Concepts
Isolate VariableEquation SolvingAlgebraic Manipulation
Isolate Variable
Isolating a variable is a fundamental skill in algebra, essential for solving equations effectively. It involves manipulating an equation to get the variable on one side, with a numeral (or an expression without the variable) on the other. This process makes it clearer what the variable represents.
For example, in the given exercise \(\Delta = -x\), our objective is to have \(x\) by itself on one side. Why is this important? Because it's the first step to discovering the value of \(x\). To accomplish this, we use the multiplication property of equality, which we'll expand upon in another section. Simply put, we multiply or divide both sides of an equation by the same number to ensure the equation remains balanced, thus isolating the variable. Here, multiplying by -1 does the trick, transforming the equation to \(x = -\Delta\).
Grasping the concept of isolating a variable is like finding the keystone in an arch; once removed, everything else falls into place, making it easer to solve for the variable.
For example, in the given exercise \(\Delta = -x\), our objective is to have \(x\) by itself on one side. Why is this important? Because it's the first step to discovering the value of \(x\). To accomplish this, we use the multiplication property of equality, which we'll expand upon in another section. Simply put, we multiply or divide both sides of an equation by the same number to ensure the equation remains balanced, thus isolating the variable. Here, multiplying by -1 does the trick, transforming the equation to \(x = -\Delta\).
Grasping the concept of isolating a variable is like finding the keystone in an arch; once removed, everything else falls into place, making it easer to solve for the variable.
Equation Solving
Solving an equation is like unraveling a mystery - you need to find the value of the unknown that makes the equation true. In algebra, an equation is a statement that two expressions are equal, and our mission is to discover the values for the variables that satisfy that equality.
The essence of solving any algebraic equation lies in isolating the variable, as we've seen in our exercise where we determined \(x = -\Delta\). But that's just the climax of the story; the plot involves various moves - applying properties like the multiplication property of equality, addition property of equality, and carrying out operations that reverse the original ones applied to the variable.
Equation solving is a cornerstone of mathematics. From simple linear equations with a single solution to complex systems of equations with multiple variables, the principles remain anchored in performing legal moves, steps that maintain the integrity of the initial equation, ultimately leading to the solution.
The essence of solving any algebraic equation lies in isolating the variable, as we've seen in our exercise where we determined \(x = -\Delta\). But that's just the climax of the story; the plot involves various moves - applying properties like the multiplication property of equality, addition property of equality, and carrying out operations that reverse the original ones applied to the variable.
Equation solving is a cornerstone of mathematics. From simple linear equations with a single solution to complex systems of equations with multiple variables, the principles remain anchored in performing legal moves, steps that maintain the integrity of the initial equation, ultimately leading to the solution.
Algebraic Manipulation
Algebraic manipulation is the art of transforming equations into simpler, more solvable forms using an ensemble of mathematical operations and properties. The key is to perform these operations without changing the fundamental nature of the equation - its solutions must remain intact.
In the context of our exercise, we executed a simple yet powerful form of algebraic manipulation. We wielded the multiplication property of equality as our tool - akin to a mathematical scalpel - to neatly excise the negative sign attached to \(x\). This property states that if you multiply (or divide) both sides of an equation by the same nonzero number, the equation's balance is preserved.
Mastering algebraic manipulation is akin to learning a new language; it's not just about knowing the grammar rules (mathematical properties) but also about fluency and finesse - the ability to discern which strategic moves will simplify an equation and lead you to the solution with elegance and efficiency.
In the context of our exercise, we executed a simple yet powerful form of algebraic manipulation. We wielded the multiplication property of equality as our tool - akin to a mathematical scalpel - to neatly excise the negative sign attached to \(x\). This property states that if you multiply (or divide) both sides of an equation by the same nonzero number, the equation's balance is preserved.
Mastering algebraic manipulation is akin to learning a new language; it's not just about knowing the grammar rules (mathematical properties) but also about fluency and finesse - the ability to discern which strategic moves will simplify an equation and lead you to the solution with elegance and efficiency.
Other exercises in this chapter
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