Solving Equations by using Addition/ Subtraction: More than Two Terms

To solve this equation for $x$, we will follow these steps:

• Simplify both sides of the equation by combining like terms.
• Rearrange the equation to isolate the variable $x$.
• Analyze the resultant equation to determine the solution.

Let's break it down:

Step 1: Simplify the left side:
The left side of the equation is $-7 + 3x - 8x$. Combine the like terms $3x$ and $-8x$:
$-7 + (3x - 8x) = -7 - 5x$

Step 2: Simplify the right side:
The right side of the equation is $9 + 3 - 5x$. Combine the constant terms $9$ and $3$:
$(9 + 3) - 5x = 12 - 5x$

Step 3: Set the simplified equation:
Now the equation is:
$-7 - 5x = 12 - 5x$

Step 4: Analyze the equation:
If we attempt to isolate $x$ by adding $5x$ to both sides, we get:
$-7 = 12$

This statement is false. Since the manipulation leads to a false statement without any variable $x$, the original equation has no solution.

Therefore, the equation cannot be true for any real number value of $x$. Thus, the correct answer is: no solution.

To solve the given problem, we'll proceed as follows:

• Step 1: Simplify both sides of the equation.
• Step 2: Check if $x$ can be isolated or analyze if the equation results in contradictions.

Now, let's work through each step:
Step 1: Simplify the left side: $x + 3 - 4x = (1x - 4x) + 3 = -3x + 3$.
Step 2: Simplify the right side: $5x + 6 - 1 - 8x = (5x - 8x) + (6 - 1) = -3x + 5$.

The simplified equation becomes:

$-3x + 3 = -3x + 5$

To solve for $x$, we attempt to isolate $x$. If we add $3x$ to both sides to eliminate the $-3x$ terms, we get:

$3 = 5$

This results in a contradiction, as 3 is not equal to 5, indicating that there is no value of $x$ that can satisfy this equation.

Therefore, the solution to the problem is no solution as indicated by the contradiction.

To solve this exercise, we first need to identify that we have an equation with an unknown,

To solve such equations, the first step will be to arrange the equation so that on one side we have the numbers and on the other side the unknowns.

$2X+7-5X-12=-8X+3$

First, we'll move all unknowns to one side.
It's important to remember that when moving terms, the sign of the number changes (from negative to positive or vice versa).

$2X+7-5X-12+8X=3$

Now we'll do the same thing with the regular numbers.

$2X-5X+8X=3-7+12$

In the next step, we'll calculate the numbers according to the addition and subtraction signs.

$2X-5X=-3X$
$-3X+8X=5X$

$3-7=-4$$$
$-4+12=8$

$5X=8$

At this stage, we want to get to a state where we have only one $X$, not $5X$,
so we'll divide both sides of the equation by the coefficient of the unknown (in this case - 5).

$X={8\over5}$

To solve this problem, we'll follow these steps:

• Simplify the term $2y \cdot \frac{1}{y}$
• Rearrange the equation to group similar terms
• Solve for $y$

Now, let's work through each step:

Step 1: Simplify the expression $2y \cdot \frac{1}{y}$.

The term $2y \cdot \frac{1}{y}$ simplifies directly to $2$ since $y$ in the numerator and denominator cancel each other out assuming $y \neq 0$. Therefore, the equation becomes:

$2 - y + 4 = 8y$

Step 2: Combine like terms on the left-hand side:

$2 + 4 = 6$, so the equation now is $6 - y = 8y$.

Step 3: Rearrange the equation to isolate $y$ on one side. Add $y$ to both sides to get rid of the negative $y$:

$6 = 8y + y$

This simplifies to:

$6 = 9y$

Step 4: Solve for $y$ by dividing both sides by 9:

$y = \frac{6}{9}$

Simplify the fraction to get:

$y = \frac{2}{3}$

Therefore, the solution to the problem is $\frac{2}{3}$.