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Algebra 1: A Comprehensive Guide to Solving Equations

 I. Solving Equations

Linear Equations

What is a linear equation?

A linear equation is an equation of the form $ax + b = y$, where $a$ and $b$ are constants and $x$ and $y$ are variables.

How to solve a linear equation using inverse operations

To solve a linear equation using inverse operations, we need to isolate the variable $x$. We can do this by performing the opposite operations that were done to $x$ on the other side of the equation.

For example, to solve the equation $2x + 3 = 7$, we would first subtract $3$ from both sides to get $2x = 4$. Then, we would divide both sides by $2$ to get $x = 2$.

How to solve a linear equation with variables on both sides

To solve a linear equation with variables on both sides, we need to move all of the terms with $x$ to one side of the equation and all of the constant terms to the other side of the equation. Then, we can solve for $x$ using the same steps that we would use to solve a linear equation with the variable on one side.

For example, to solve the equation $3x – 2 = 2x + 5$, we would first subtract $2x$ from both sides to get $x – 2 = 5$. Then, we would add $2$ to both sides to get $x = 7$.

How to solve a linear equation with parentheses

To solve a linear equation with parentheses, we need to distribute the parentheses on both sides of the equation. Then, we can solve for $x$ using the same steps that we would use to solve a linear equation without parentheses.

For example, to solve the equation $2(x + 3) = 7$, we would first distribute the parentheses to get $2x + 6 = 7$. Then, we would subtract $6$ from both sides to get $2x = 1$. Finally, we would divide both sides by $2$ to get $x = \dfrac{1}{2}$.

Quadratic Equations

What is a quadratic equation?

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

How to solve a quadratic equation by factoring

To solve a quadratic equation by factoring, we need to factor the quadratic trinomial on the left side of the equation. Once we have factored the trinomial, we can set each factor equal to zero and solve for $x$.

For example, to solve the quadratic equation $x^2 + 2x – 3 = 0$, we would first factor the trinomial to get $(x + 3)(x – 1) = 0$. Then, we would set each factor equal to zero and solve for $x$:

  • $x + 3 = 0$
  • $x – 1 = 0$

Solving the first equation, we get $x = -3$. Solving the second equation, we get $x = 1$. Therefore, the solutions to the quadratic equation $x^2 + 2x – 3 = 0$ are $x = -3$ and $x = 1$. How to solve a quadratic equation using the quadratic formula

The quadratic formula is a formula that can be used to solve any quadratic equation. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

To use the quadratic formula, we simply substitute the values of $a$, $b$, and $c$ into the formula and solve for $x$.

For example, to solve the quadratic equation $x^2 + 2x – 3 = 0$, we would substitute the values of $a = 1$, $b = 2$, and $c = -3$ into the quadratic formula:

x = (-2 ± √(2² - 4 × 1 × -3)) / 2 × 1
x = (-2 ± √(16)) / 2
x = (-2 ± 4) / 2

Therefore, the solutions to the quadratic equation $x^2 + 2x – 3 = 0$ are $x = -3$ and $x = 1$.

Systems of Equations

What is a system of equations?

A system of equations is a set of two or more equations that have the same variables.

How to solve a system of equations by elimination

To solve a system of equations by elimination, we need to eliminate one of the variables by adding or subtracting the equations. This can be done by finding two equations with opposite coefficients for the variable we want to eliminate.

For example, to solve the system of equations:

2x + y = 5
x - y = -1

We would first add the two equations together to get $3x = 4$. Then, we would divide both sides of the equation by $3$ to get $x = \dfrac{4}{3}$.

Now that we know the value of $x$, we can substitute it into one of the original equations to solve for $y$. For example, substituting $x = \dfrac{4}{3}$ into the first equation, we get:

2(4/3) + y = 5
8/3 + y = 5
y = 5 - 8/3
y = 1/3

Therefore, the solution to the system of equations is $x = \dfrac{4}{3}$ and $y = \dfrac{1}{3}$.

How to solve a system of equations by substitution

To solve a system of equations by substitution, we need to solve one of the equations for one of the variables and substitute that value into the other equation. Then, we can solve for the other variable.

For example, to solve the system of equations:

2x + y = 5
x - y = -1

We would first solve the first equation for $y$:

y = 5 - 2x

Then, we would substitute this value of $y$ into the second equation:

x - (5 - 2x) = -1
x - 5 + 2x = -1
3x - 5 = -1
3x = 4
x = 4/3

Now that we know the value of $x$, we can substitute it into either of the original equations to solve for $y$. For example, substituting $x = \dfrac{4}{3}$ into the first equation, we get:

2(4/3) + y = 5
8/3 + y = 5
y = 5 - 8/3
y = 1/3

Therefore, the solution to the system of equations is $x = \dfrac{4}{3}$ and $y = \dfrac{1}{3}$.

Inequalities

What is an inequality?

An inequality is a mathematical statement that compares two expressions. Inequalities can be used to represent relationships between quantities, such as “greater than,” “less than,” or “equal to.”

How to solve a linear inequality

To solve a linear inequality, we need to isolate the variable on one side of the inequality and put all of the constant terms on the other side of the inequality. Then, we can compare the coefficients of the variable on both sides of the inequality and determine the solution.

For example, to solve the linear inequality $x + 3 > 5$, we would first subtract $3$ from both sides to get $x > 2$. Therefore, the solution to the inequality is $x > 2$.

How to solve a quadratic inequality

To solve a quadratic inequality, we need to graph the quadratic function and determine the regions where the graph is above or below the $x$-axis. The regions where the graph is above the $x$-axis are the solutions to the inequality.

For example, to solve the quadratic inequality $x^2 + 2x – 3 > 0$, we would first graph the quadratic function $f(x) = x^2 + 2x – 3$. The graph of the function is shown below.

[Image of a quadratic function graph]

The graph is above the $x$-axis between the values of $x = -3$ and $x = 1$. Therefore, the solution to the inequality is $x > -3$ or $x < 1$.

Conclusion

This guide has provided an overview of Algebra 1 concepts, along with practice questions and helpful tips. By mastering the concepts covered in this guide, students will be well-prepared for their future math studies.

FAQs

What is the difference between an equation and an inequality?

An equation is a mathematical statement that shows that two expressions are equal. An inequality is a mathematical statement that compares two expressions.

How do I solve a quadratic equation?

There are three ways to solve a quadratic equation: by factoring, by completing the square, and by using the quadratic formula.

How do I solve a system of equations?

There are two ways to solve a system of equations: by elimination and by substitution.

What is the difference between a linear function and a quadratic function?

A linear function is a function whose graph is a straight line. A quadratic function is a function whose graph is a parabola.

How do I factor a polynomial?

There are a number of different ways to factor a polynomial. Some common methods include factoring by grouping and factoring by the sum-product pattern.

What is the difference between a rational exponent and a complex number?

A rational exponent is an exponent that is a fraction. A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

 

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