Linear Equations Chapter 4

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Questions and Answers

What is the general form of a linear equation in two variables?

The general form of a linear equation in two variables is $Ax + By + C = 0$, where A, B, and C are constants.

How can you determine if a linear equation has infinite solutions?

A linear equation has infinite solutions if it can be rewritten in the form $y = mx + b$ where $m$ and $b$ are both equal for two different equations.

What does an ordered pair (x, y) represent in the context of linear equations?

An ordered pair (x, y) represents a specific solution to a linear equation, indicating the values of x and y that satisfy the equation.

How can you graph a linear equation in two variables?

To graph a linear equation, you can find two or more points that satisfy the equation and plot them on the Cartesian plane, then draw a straight line through these points. Signup and view all the answers

Give an example of a real-world application of linear equations.

An example of a real-world application is determining the total cost of items based on quantity, represented by the equation $C = px$, where $C$ is total cost, $p$ is price per unit, and $x$ is quantity. Signup and view all the answers

What happens to the solution of a linear equation if you multiply both sides by a non-zero number?

Multiplying both sides of a linear equation by a non-zero number does not change its solution; it remains equivalent. Signup and view all the answers

What does it mean if two linear equations intersect at a point on the Cartesian plane?

If two linear equations intersect at a point, that point represents the unique solution where both equations are satisfied. Signup and view all the answers

How would you express the total score of two players in a cricket match using a linear equation?

The total score of two players can be expressed as $x + y = 176$, where $x$ and $y$ are the scores of the two players. Signup and view all the answers

Convert the equation 2x + 3y = 9.35 into standard form ax + by + c = 0 and identify the values of a, b, and c.

The standard form is 2x + 3y - 9.35 = 0, where a = 2, b = 3, and c = -9.35. Signup and view all the answers

How many solutions exist for a linear equation in two variables, and what does a solution represent?

A linear equation in two variables has infinitely many solutions represented as ordered pairs (x, y). Signup and view all the answers

Given the equation 2x + 3y = 12, provide two examples of ordered pairs that are solutions.

The ordered pairs (3, 2) and (0, 4) are solutions for the equation 2x + 3y = 12. Signup and view all the answers

Explain how to generate other solutions for the equation 2x + 3y = 12.

By choosing a value for x, one can rearrange the equation to solve for y, generating different ordered pairs. Signup and view all the answers

Is the ordered pair (1, 4) a solution for the equation 2x + 3y = 12? Justify your answer.

No, (1, 4) is not a solution because substituting gives 2(1) + 3(4) = 14, not 12. Signup and view all the answers

What does the infinite number of solutions mean in the context of graphing linear equations?

It means that the corresponding graph is a straight line where every point represents a solution to the equation. Signup and view all the answers

Describe the significance of finding multiple solutions for the equation 2x + 3y = 12 in real-world applications.

Multiple solutions indicate various combinations of values that satisfy a constraint, such as budget or resource allocation. Signup and view all the answers

Convert the equation x = 3y into the standard form ax + by + c = 0 and define the values of a, b, and c.

The standard form is x - 3y = 0, where a = 1, b = -3, and c = 0. Signup and view all the answers

What is a linear equation in two variables, and can you provide an example?

A linear equation in two variables is of the form $ax + by + c = 0$. An example is $2x + 3y - 4 = 0$. Signup and view all the answers

If $x + y = 176$, what can you say about the set of ordered pairs $(x, y)$ that satisfy this equation?

The set of ordered pairs $(x, y)$ that satisfy the equation represent all combinations of $x$ and $y$ that sum to 176. Signup and view all the answers

When graphed, what does the graph of a linear equation indicate about the possible solutions?

The graph of a linear equation shows a straight line where each point on the line represents a solution to the equation. Signup and view all the answers

In the context of a linear equation, what does it mean if the equation has infinite solutions?

An equation has infinite solutions when every point on a line satisfies the equation, indicating algebraic dependence between variables. Signup and view all the answers

How can the statement 'the cost of a notebook is twice the cost of a pen' be expressed as a linear equation in two variables?

Let the cost of a pen be $p$ and the cost of a notebook be $n$, the equation can be written as $n - 2p = 0$. Signup and view all the answers

What is the value of $a$, $b$, and $c$ in the equation $5x - 3y - 4 = 0$?

In the equation $5x - 3y - 4 = 0$, $a = 5$, $b = -3$, and $c = -4$. Signup and view all the answers

How would you convert the equation $2x = y$ into standard form?

The equation $2x = y$ can be rewritten as $2x - y = 0$. Signup and view all the answers

In what way can the linear equation $x - 3y - 4 = 0$ be used to solve for $y$ in terms of $x$?

You can rearrange it to express $y$ as $y = (1/3)x - (4/3)$. Signup and view all the answers

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Study Notes

Introduction to Linear Equations in Two Variables

Linear equations in one variable have unique solutions, such as x + 1 = 0.
Transitioning to two variables introduces pairs of solutions, one for each variable (x, y).
Questions arise regarding the existence and uniqueness of solutions for two-variable equations.

Understanding Linear Equations

Example equation: 2x + 5 = 0; solution is x = -5/2.
Solutions remain consistent when the same number is added/subtracted or when both sides are multiplied/divided by a non-zero number.
Real-life example: In cricket, two players score a combined total, leading to a linear equation x + y = 176.

Forms of Linear Equations

Standard form: ax + by + c = 0, with real number coefficients a and b (not both zero).
Examples of different linear equations in two variables:
- 2s + 3t = 5
- p + 4q = 7
- πu + 5v = 9
- 2x - 7y = 3

Solutions of Linear Equations

A linear equation in two variables has multiple solutions; represented as ordered pairs (x, y).
Example with specific solutions:
- For 2x + 3y = 12, valid solutions include (3, 2), (0, 4), and (6, 0).
Solutions can be derived by substituting a value for one variable and solving for the other, showcasing infinite possibilities.

Expression and Transformation of Equations

Each linear equation can be transformed into the standard form.
Examples:
- 2x + 3y = 4.37 transforms to 2x + 3y - 4.37 = 0.
- x - 4 = 3y becomes x - 3y - 4 = 0.
Various forms of equations still represent linear relationships, aiding in understanding their structure.

Identifying Variables in Equations

Equations such as ax + b = 0 can be represented in two variables, e.g., 4 - 3x = 0 as -3x + 0y + 4 = 0.
Specific equations like x = -5 or y = 2 can be expressed as:
- 1.x + 0.y + 5 = 0
- 0.x + 1.y - 2 = 0

Exercise Example

To represent the cost of a notebook being twice that of a pen, formulate a linear equation such as x = 2y, where x is the cost of a notebook and y is the cost of a pen.

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