Determine The Value Of The Variable That Makes Each Equation True.$\[ \begin{array}{|c|c|} \hline \text{Expression} & \text{Value} \\ \hline a \cdot 3 = -30 & \\ \hline -9 \cdot B = -45 & \\ \hline -89 \cdot 12 = C & \\ \hline d \cdot 88 = -88000 &

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In mathematics, equations are used to represent relationships between variables and constants. Solving equations involves finding the value of the variable that makes the equation true. In this article, we will determine the value of the variable that makes each of the given equations true.

Solving the First Equation

The first equation is a⋅3=−30a \cdot 3 = -30. To solve for aa, we need to isolate the variable aa on one side of the equation.

Step 1: Divide both sides of the equation by 3

To isolate aa, we need to get rid of the 3 that is being multiplied by aa. We can do this by dividing both sides of the equation by 3.

a⋅33=−303\frac{a \cdot 3}{3} = \frac{-30}{3}

Step 2: Simplify the equation

When we divide both sides of the equation by 3, the 3 on the left-hand side cancels out, leaving us with just aa.

a=−303a = \frac{-30}{3}

Step 3: Simplify the fraction

To simplify the fraction, we can divide the numerator and denominator by their greatest common divisor, which is 3.

a=−101a = \frac{-10}{1}

Step 4: Write the final answer

The final answer is a=−10a = -10.

Solving the Second Equation

The second equation is −9⋅b=−45-9 \cdot b = -45. To solve for bb, we need to isolate the variable bb on one side of the equation.

Step 1: Divide both sides of the equation by -9

To isolate bb, we need to get rid of the -9 that is being multiplied by bb. We can do this by dividing both sides of the equation by -9.

−9⋅b−9=−45−9\frac{-9 \cdot b}{-9} = \frac{-45}{-9}

Step 2: Simplify the equation

When we divide both sides of the equation by -9, the -9 on the left-hand side cancels out, leaving us with just bb.

b=−45−9b = \frac{-45}{-9}

Step 3: Simplify the fraction

To simplify the fraction, we can divide the numerator and denominator by their greatest common divisor, which is 9.

b=−5−1b = \frac{-5}{-1}

Step 4: Write the final answer

The final answer is b=5b = 5.

Solving the Third Equation

The third equation is −89⋅12=c-89 \cdot 12 = c. To solve for cc, we need to isolate the variable cc on one side of the equation.

Step 1: Multiply both sides of the equation by -1

To isolate cc, we need to get rid of the -89 that is being multiplied by 12. We can do this by multiplying both sides of the equation by -1.

−1⋅(−89⋅12)=−1⋅c-1 \cdot (-89 \cdot 12) = -1 \cdot c

Step 2: Simplify the equation

When we multiply both sides of the equation by -1, the -1 on the left-hand side cancels out, leaving us with just cc.

c=89⋅12c = 89 \cdot 12

Step 3: Multiply the numbers

To find the value of cc, we need to multiply 89 and 12.

c=1068c = 1068

Step 4: Write the final answer

The final answer is c=1068c = 1068.

Solving the Fourth Equation

The fourth equation is d⋅88=−88000d \cdot 88 = -88000. To solve for dd, we need to isolate the variable dd on one side of the equation.

Step 1: Divide both sides of the equation by 88

To isolate dd, we need to get rid of the 88 that is being multiplied by dd. We can do this by dividing both sides of the equation by 88.

d⋅8888=−8800088\frac{d \cdot 88}{88} = \frac{-88000}{88}

Step 2: Simplify the equation

When we divide both sides of the equation by 88, the 88 on the left-hand side cancels out, leaving us with just dd.

d=−8800088d = \frac{-88000}{88}

Step 3: Simplify the fraction

To simplify the fraction, we can divide the numerator and denominator by their greatest common divisor, which is 8.

d=−1100011d = \frac{-11000}{11}

Step 4: Write the final answer

The final answer is d=−1000d = -1000.

In conclusion, we have solved four equations to find the value of the variable that makes each equation true. The final answers are a=−10a = -10, b=5b = 5, c=1068c = 1068, and d=−1000d = -1000.

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In the previous article, we solved four equations to find the value of the variable that makes each equation true. In this article, we will answer some frequently asked questions (FAQs) about solving equations.

Q: What is an equation?

A: An equation is a statement that says two things are equal. It is a mathematical statement that contains an equal sign (=) and is used to represent a relationship between variables and constants.

Q: What is a variable?

A: A variable is a letter or symbol that represents a value that can change. In an equation, the variable is the value that we are trying to find.

Q: How do I solve an equation?

A: To solve an equation, we need to isolate the variable on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an equation. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an equation?

A: To simplify an equation, we need to combine like terms and eliminate any unnecessary operations. This can be done by adding or subtracting the same value to both sides of the equation.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation 2x + 3 = 5 is a linear equation.

Q: What is a quadratic equation?

A: A quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously. For example, the system of equations:

x + y = 2 x - y = 1

can be solved using substitution or elimination methods.

Q: How do I solve a system of equations?

A: To solve a system of equations, we can use substitution or elimination methods. Substitution involves solving one equation for one variable and then substituting that value into the other equation. Elimination involves adding or subtracting the two equations to eliminate one variable.

Q: What is a graphing calculator?

A: A graphing calculator is a calculator that can graph equations and functions. It is a useful tool for visualizing the behavior of equations and functions.

Q: How do I use a graphing calculator?

A: To use a graphing calculator, we need to enter the equation or function that we want to graph. We can then use the calculator's built-in functions to graph the equation or function.

In conclusion, solving equations is an important part of mathematics and is used in a wide range of applications. By understanding the basics of solving equations, we can solve a wide range of problems and make informed decisions.