Create Expressions and Solve Linear Equations, Including Fractional Expressions

Introduction

Key Concepts

1. Understanding Expressions and Equations

In algebra, an expression is a combination of numbers, variables, and mathematical operations (such as addition and multiplication) that represents a particular value. For example, $3x + 2$ is an expression where $x$ is a variable.

An equation, on the other hand, states that two expressions are equal. It includes an equality sign (=). For example, $3x + 2 = 11$ is an equation.

2. Creating Linear Expressions

Creating linear expressions involves identifying the relationship between variables and constants. A linear expression in one variable has the form $ax + b$, where $a$ and $b$ are constants.

**Example:** If the cost $C$ of buying $x$ notebooks is $2x + 5$, then $C = 2x + 5$ is a linear expression.

3. Solving Linear Equations

Solving a linear equation involves finding the value of the variable that makes the equation true. The goal is to isolate the variable on one side of the equation.

**Steps to Solve:**

1. Simplify both sides of the equation if necessary.
2. Move all terms containing the variable to one side and constants to the other side.
3. Combine like terms.
4. Divide or multiply to solve for the variable.

**Example:** Solve $3x + 2 = 11$.

**Solution:**

1. Subtract 2 from both sides: $3x = 9$
2. Divide both sides by 3: $x = 3$

4. Fractional Expressions

Fractional expressions involve variables in the numerator, denominator, or both. Solving equations with fractional expressions requires careful handling to eliminate the fractions.

**Example:** Solve $\frac{2x + 3}{4} = 5$.

**Solution:**

1. Multiply both sides by 4 to eliminate the denominator: $2x + 3 = 20$
2. Subtract 3 from both sides: $2x = 17$
3. Divide by 2: $x = \frac{17}{2}$ or $x = 8.5$

5. Applications of Linear Equations

Linear equations are used to model real-life situations such as calculating costs, distances, and other quantities that change at a constant rate.

**Real-World Example:** If a taxi charges a base fare of $3 plus $2 per mile, the total cost $C$ for $m$ miles can be expressed as $C = 2m + 3$. To find out how many miles can be traveled with $11, set $2m + 3 = 11$ and solve for $m$.

6. Graphing Linear Equations

Graphing linear equations involves plotting the equation on a coordinate plane to visualize the relationship between variables.

**Steps to Graph:**

1. Rewrite the equation in slope-intercept form: $y = mx + c$.
2. Identify the slope ($m$) and y-intercept ($c$).
3. Plot the y-intercept on the y-axis.
4. Use the slope to find another point.
5. Draw the line through the plotted points.

**Example:** Graph $y = 2x + 1$.

**Solution:**

1. The slope $m$ is 2, and the y-intercept $c$ is 1.
2. Plot the point (0, 1) on the y-axis.
3. From (0, 1), use the slope to rise 2 units and run 1 unit to the right, reaching (1, 3).
4. Draw a line through (0, 1) and (1, 3).

7. Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. Solving the system involves finding the values of the variables that satisfy all equations simultaneously.

**Methods to Solve:**

• Graphical Method: Plot both equations on the same graph and identify the intersection point.
• Substitution Method: Solve one equation for one variable and substitute it into the other equation.
• Elimination Method: Add or subtract equations to eliminate one variable, then solve for the remaining variable.

**Example:** Solve the system: $$ \begin{align*} 2x + y &= 10 \\ x - y &= 2 \end{align*} $$

**Solution (Elimination Method):**

1. Add both equations: $(2x + y) + (x - y) = 10 + 2$ ⟹ $3x = 12$
2. Divide by 3: $x = 4$
3. Substitute $x = 4$ into the second equation: $4 - y = 2$ ⟹ $y = 2$

Therefore, the solution is $x = 4$ and $y = 2$.

8. Word Problems Involving Linear Equations

Word problems require translating a real-world scenario into a linear equation or system of equations.

**Example:** Sarah buys 3 pens and 2 notebooks for a total of $11. If a pen costs $2, find the cost of a notebook.

**Solution:**

1. Let the cost of a notebook be $n$.
2. The total cost equation: $3(2) + 2n = 11$
3. Simplify: $6 + 2n = 11$
4. Subtract 6: $2n = 5$
5. Divide by 2: $n = \frac{5}{2}$ or $n = 2.5$

Advanced Concepts

1. Solving Equations with Variables on Both Sides

Some linear equations have variables on both sides. Solving such equations requires collecting like terms on one side.

**Example:** Solve $2x + 3 = x + 7$.

**Solution:**

1. Subtract $x$ from both sides: $x + 3 = 7$
2. Subtract 3: $x = 4$

2. Equations with Fractional Expressions

Solving equations that involve fractions requires eliminating the denominators to simplify the equation.

**Example:** Solve $\frac{3x - 2}{5} = \frac{x + 1}{2}$.

**Solution:**

1. Find the least common denominator (LCD) of 5 and 2, which is 10.
2. Multiply both sides by 10: $10 \cdot \frac{3x - 2}{5} = 10 \cdot \frac{x + 1}{2}$ ⟹ $2(3x - 2) = 5(x + 1)$
3. Expand both sides: $6x - 4 = 5x + 5$
4. Subtract $5x$ from both sides: $x - 4 = 5$
5. Add 4: $x = 9$

3. Systems Involving Fractions

When systems of equations include fractions, it's essential to eliminate the denominators first to simplify the equations.

**Example:** Solve the system: $$ \begin{align*} \frac{x}{2} + \frac{y}{3} &= 5 \\ \frac{2x}{3} - \frac{y}{4} &= 3 \end{align*} $$

**Solution:**

1. Find the LCD for each equation.
2. Multiply the first equation by 6: $3x + 2y = 30$
3. Multiply the second equation by 12: $8x - 3y = 36$
4. Now, solve the system:
   • Multiply the first equation by 3: $9x + 6y = 90$
   • Multiply the second equation by 2: $16x - 6y = 72$
   • Add both equations: $25x = 162$ ⟹ $x = \frac{162}{25} = 6.48$
   • Substitute $x$ into the first simplified equation: $3(6.48) + 2y = 30$ ⟹ $19.44 + 2y = 30$ ⟹ $2y = 10.56$ ⟹ $y = 5.28$

4. Applications in Real-World Contexts

Advanced applications involve complex real-world scenarios where multiple linear equations model various aspects of the problem.

**Example:** A company sells two products, A and B. The profit from product A is $3 per unit, and from product B is $4 per unit. The total profit is $25, and the company sold a total of 9 units. Determine the number of units sold for each product.

**Solution:**

1. Let $x$ be the units of product A and $y$ be the units of product B.
2. Set up the equations:
   • Profit equation: $3x + 4y = 25$
   • Units equation: $x + y = 9$
3. Solve the system using substitution or elimination.
4. Using substitution:
   • From the units equation: $x = 9 - y$
   • Substitute into the profit equation: $3(9 - y) + 4y = 25$ ⟹ $27 - 3y + 4y = 25$
   • Simplify: $27 + y = 25$ ⟹ $y = -2$
5. The negative value indicates no solution in this context, suggesting an error in the problem's parameters.

**Conclusion:** No combination of 9 units sold between product A and B yields a profit of $25 based on the given prices.

5. Equations with Absolute Values

Absolute value equations involve expressions within absolute value symbols. Solving these requires considering both positive and negative scenarios.

**Example:** Solve $|2x - 3| = 7$.

**Solution:**

1. Set up two equations:
   • $2x - 3 = 7$
   • $2x - 3 = -7$
2. Solving the first equation: $2x = 10$ ⟹ $x = 5$
3. Solving the second equation: $2x = -4$ ⟹ $x = -2$

Therefore, $x = 5$ or $x = -2$.

6. Inequalities and Linear Equations

While focusing on linear equations, it's beneficial to understand their relation to inequalities, which express a range of solutions.

**Example:** Solve $3x - 2 > 7$.

**Solution:**

1. Add 2 to both sides: $3x > 9$
2. Divide by 3: $x > 3$

Thus, the solution is all real numbers greater than 3.

7. Linear Equations in Two Variables

Equations in two variables represent lines on a coordinate plane. Understanding their properties is crucial for interpreting graphical solutions.

**Standard Form:** $Ax + By = C$.

**Slope-Intercept Form:** $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

**Parallel and Perpendicular Lines:**

• Parallel Lines: Have equal slopes ($m_1 = m_2$).
• Perpendicular Lines: Slopes are negative reciprocals ($m_1 = -\frac{1}{m_2}$).

8. Parametric Equations

Parametric equations express variables as functions of a third variable, often time. They are useful in modeling motion and other dynamic systems.

**Example:** A line can be described parametrically as: $$ \begin{align*} x &= 2 + 3t \\ y &= 4 - t \end{align*} $$

Here, $t$ is the parameter that varies over the real numbers.

9. Linear Programming

Linear programming involves optimizing (maximizing or minimizing) a linear objective function subject to linear constraints. It has applications in various fields like economics, engineering, and logistics.

**Example:** Maximize profit $P = 3x + 4y$ subject to: $$ \begin{align*} 2x + y &\leq 20 \\ x + 2y &\leq 20 \\ x &\geq 0, y \geq 0 \end{align*} $$

**Solution:**

1. Graph the constraints to find the feasible region.
2. Identify the corner points of the feasible region.
3. Evaluate the objective function at each corner point.
4. The maximum profit occurs at the point where $P$ is highest.

10. Introduction to Matrix Methods

Matrix methods provide a systematic way to solve systems of linear equations, especially those with three or more variables.

**Example:** Solve the system using matrices: $$ \begin{align*} x + y + z &= 6 \\ 2x + 3y + z &= 14 \\ x + y + 2z &= 9 \end{align*} $$

**Solution:**

1. Write the augmented matrix: $$ \begin{bmatrix} 1 & 1 & 1 & | & 6 \\ 2 & 3 & 1 & | & 14 \\ 1 & 1 & 2 & | & 9 \end{bmatrix} $$
2. Use row operations to reduce the matrix to row-echelon form.
3. Back-substitute to find the values of $x$, $y$, and $z$.

After performing the operations, the solution is $x = 2$, $y = 3$, and $z = 1$.

11. Exploring Functions and Their Representations

Linear equations can represent functions, establishing a direct relationship between variables. Understanding different representations (graphical, numerical, algebraic) is crucial.

**Example:** The function $f(x) = 2x + 3$ can be represented as:

• Algebraic: $f(x) = 2x + 3$
• Graphical: A straight line with slope 2 and y-intercept 3.
• Numerical: A table of values showing corresponding $x$ and $f(x)$.

12. Solving Linear Equations with Multiple Variables

Equations with more than one variable require additional equations to find unique solutions. Methods include substitution, elimination, and matrix techniques.

**Example:** Solve: $$ \begin{align*} x + 2y - z &= 4 \\ 2x - y + 3z &= 10 \\ -x + \frac{1}{2}y - z &= -2 \end{align*} $$

**Solution:**

1. Use elimination to reduce the system step by step.
2. Eventually, find the values of $x$, $y$, and $z$ that satisfy all equations.

The solution is $x = 2$, $y = 3$, and $z = 1$.

13. Understanding Linear Inequalities

Linear inequalities express ranges of possible solutions rather than specific values. They are often represented graphically using shaded regions.

**Example:** Solve and graph $2x - 3

**Solution:**

1. Add 3 to both sides: $2x
2. Divide by 2: $x

**Graph:** A number line with an open circle at 5 and shading to the left, indicating all real numbers less than 5.

14. Parametric Solutions to Systems

When a system has infinitely many solutions, parametric equations describe the set of all possible solutions using one or more parameters.

**Example:** Solve the system: $$ \begin{align*} x + y &= 4 \\ 2x + 2y &= 8 \end{align*} $$

**Solution:** The second equation is a multiple of the first, indicating infinitely many solutions. Let $x = t$, then $y = 4 - t$, where $t$ is a parameter.

15. Linear Equations in Higher Dimensions

Extending linear equations to three or more variables involves working in higher-dimensional spaces, which can be visualized using hyperplanes.

**Example:** A plane in three-dimensional space can be represented by an equation like $x + 2y + 3z = 6$.

Understanding the geometry of such equations aids in comprehending more complex systems and their intersections.

Comparison Table

Summary and Key Takeaways