Solving Equations Using All Methods - Examples, Exercises and Solutions

First we will move terms so that -b remains remains on the left side of the equation.

We'll move 8 to the right-hand side, making sure to retain the plus and minus signs accordingly:

$-b=6-8$

Then we will subtract as follows:

$-b=-2$

Finally, we will divide both sides by -1 (be careful with the plus and minus signs when dividing by a negative):

$\frac{-b}{-1}=\frac{-2}{-1}$

$b=2$

Let's combine all the x terms together:

$7x+4x+5x=11x+5x=16x$

The resulting equation is:

$16x=0$

Now let's divide both sides by 16:

$\frac{16x}{16}=\frac{0}{16}$

$x=\frac{0}{16}=0$

Let's first expand the parentheses using the formula:

$a(x+b)=ax+ab$

$(2\times x)+(2\times4)+8=0$

$2x+8+8=0$

Next, we will substitute in our terms accordingly:

$2x+16=0$

Then, we will move the 16 to the left-hand side, keeping the appropriate sign:

$2x=-16$

Finally, we divide both sides by 2:

$\frac{2x}{2}=-\frac{16}{2}$

$x=-8$

Let's proceed to solve the linear equation $3(a+1) - 3 = 0$:

Step 1: Distribute the 3 in the expression $3(a+1)$.

We get:
$3 \cdot a + 3 \cdot 1 - 3 = 0$

This simplifies to:
$3a + 3 - 3 = 0$

Step 2: Simplify the expression by combining like terms.

We simplify this to:
$3a + 0 = 0$ or simply $3a = 0$

Step 3: Isolate $a$ by dividing both sides by 3.

$\frac{3a}{3} = \frac{0}{3}$

Thus,
$a = 0$

Therefore, the solution to the problem is $a = 0$.

The correct choice is the option corresponding to $a = 0$.

Let's solve the equation $-16 + a = -17$ by isolating the variable $a$.

To isolate $a$, add 16 to both sides of the equation to cancel out the $-16$:

$-16 + a + 16 = -17 + 16$

This simplification results in:

$a = -1$

Thus, the solution to the equation $-16 + a = -17$ is $a = -1$.

If we review the answer choices given, the correct answer is Choice 4, $-1$.

The solution to the problem is $a = -1$.

$ax=\frac{c}{b}$

$x=\frac{c}{b\cdot a}$

Step-by-step solution:

1. Begin with the equation: $x + 9 = 15$

2. Subtract 9 from both sides: $x + 9 - 9 = 15 - 9$, which simplifies to $x = 6$

The goal is to solve the equation $4 = 3y$ to find the value of $y$. To do this, we can follow these steps:

• Step 1: Divide both sides of the equation by 3 to isolate $y$.
• Step 2: Simplify the result to solve for $y$.

Now, let's work through the solution:

Step 1: We start with the equation:

$4 = 3y$

To solve for $y$, divide both sides by 3:

$y = \frac{4}{3}$

Step 2: Simplify the fraction:

$y = \frac{4}{3} = 1 \frac{1}{3}$

Therefore, the solution to the equation is $y = 1 \frac{1}{3}$.

This corresponds to choice $y = 1\frac{1}{3}$ in the provided multiple-choice answers.

To find the value of $a$, we must solve the given linear equation:

$11 = a - 16$

We aim to isolate $a$ by performing operations that maintain the balance of the equation. Currently, $a$ is being decreased by 16. To reverse this, we need to add 16 to both sides.

Step-by-step:

• Start with the given equation: $11 = a - 16$.
• Add 16 to both sides to start isolating $a$:

$11 + 16 = a - 16 + 16$

• This simplifies to:

$27 = a$

Thus, the value of $a$ is 27.

Therefore, the solution to the equation $11 = a - 16$ is $a = 27$.

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(7\times-2x)+(7\times5)=77$

We multiply accordingly

$-14x+35=77$

We will move the 35 to the right section and change the sign accordingly:

$-14x=77-35$

We solve the subtraction exercise on the right side and we will obtain:

$-14x=42$

We divide both sections by -14

$\frac{-14x}{-14}=\frac{42}{-14}$

$x=-3$

To solve for $a$, we need to isolate it on one side of the equation. Starting with:

$a-5=10$

Add $5$ to both sides to get:

$a-5+5=10+5$

This simplifies to:

$a=15$

Therefore, the solution is$a = 15$.

To solve for $b$, we need to isolate it on one side of the equation. Starting with:

$b+6=14$

Subtract$6$ from both sides to get:

$b+6-6=14-6$

This simplifies to:

$b=8$

Therefore, the solution is $b = 8$.

To solve for $x$ in the equation $6x = 72$, follow these steps:

Step 1: Identify the equation and the coefficient of $x$.
The given equation is $6x = 72$, where the coefficient of $x$ is 6.

Step 2: Isolate $x$ by dividing both sides of the equation by the coefficient (6).
Perform the division: $x = \frac{72}{6}$.

Step 3: Simplify the result.
Calculating $\frac{72}{6}$, we get $x = 12$.

Therefore, the solution to the equation is $x = 12$.

To solve for $x$, start by isolating $x$ on one side of the equation:
Subtract 3 from both sides:
$x + 3 - 3 = 7 - 3$ simplifies to
$x = 4$.

To solve for $x$, start by isolating $x$ on one side of the equation:
Subtract 7 from both sides:
$x + 7 - 7 = 12 - 7$ simplifies to
$x = 5$.