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Magazine Chapter 3: Problem 1
Solve these equations. Give reasons for each step. a. \(-x=7\) b. \(4.2=-2 x-42.6\)
Short Answer
Expert verified
(a) \(x = -7\); (b) \(x = -23.4\).
Step by step solution
01
Understanding Equation (a)
We are given the equation \(-x = 7\). This means we need to find the value of \(x\) by isolating \(x\) on one side.
02
Solving for x in Equation (a)
Since the equation is \(-x = 7\), we need to multiply both sides by \(-1\) to solve for \(x\). By doing this, we get:\[-x \times (-1) = 7 \times (-1)\] which simplifies to \[x = -7\].
03
Understanding Equation (b)
Now, consider the equation \(4.2 = -2x - 42.6\). The goal is to isolate \(x\), starting by moving all terms involving \(x\) to one side and constant terms to the other side.
04
Adding 42.6 to Both Sides
Add 42.6 to both sides of the equation to isolate the term with \(x\):\[4.2 + 42.6 = -2x - 42.6 + 42.6\] which simplifies to \[46.8 = -2x\].
05
Solve for x
To solve \(-2x = 46.8\), divide both sides by \(-2\) to isolate \(x\):\[-2x / -2 = 46.8 / -2\] resulting in \[x = -23.4\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Isolating Variables
Isolating a variable is a fundamental skill when solving equations, as it helps you find out what the unknown variable is equal to. Imagine you have an equation and your goal is to find the value of a specific variable, like finding the key to unlock a box. You want to "get the variable alone" on one side of the equation. This often means moving things around and performing operations to both sides of the equation so that the variable stands alone.
- Look at each side of the equation and decide which operations will help move the variable to just one side.
- Perform the same operations on both sides to keep the equation balanced, like using a seesaw.
Multiplying Both Sides
Sometimes, the variable is attached with a negative sign or a fraction, and multiplying both sides of an equation helps break free from this constraint. Let鈥檚 consider a negative sign in front of a variable. Multiplying the equation by e.g., \(-x = 7\)means to get \(x\) by multiplying both sides by \(-1\). This results in:\(-x \times (-1) = 7 \times (-1)\)which simplifies to \(x = -7\).
- Remember to multiply every term on both sides of the equation by the same number.
- If you multiply by a negative number, remember that it changes the sign of the terms, flipping the direction of inequalities (but only in inequalities).
Adding to Both Sides
Adding or subtracting a number to both sides of an equation is a common technique to shift terms from one side of the equation to the other. Generally, this is handy when you need to remove a constant term from alongside the variable you're solving for. For instance, consider
4.2 = -2x - 42.6,
you can get rid of the
-42.6
by
adding 42.6
to both sides:
4.2 + 42.6 = -2x - 42.6 + 42.6,
resulting in
46.8 = -2x.
- This balances things out and gets you one step closer to isolating the variable.
- The trick works for subtraction too. It reverses or neutralizes the operations on the side where the variable is.
Dividing Both Sides
Once you've isolated the variable term on one side, often by adding or multiplying, you might notice that the variable is still tied up in an equation with other numbers. This is where dividing both sides comes in handy. Say, if you have an equation
-2x = 46.8,
to find the value of
x,
you divide both sides by
-2:
(-2x)/(-2) = 46.8/(-2),
which simplifies to
x = -23.4.
- Double-check your division to ensure correct simplification and maintain balance of the equation.
- Remember, dividing by a positive or negative number affects the sign of the result, so handle direction changes with care, particularly in inequalities.
