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Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Solutions for 2x + 3y = 8
Introduction: In this problem, we're tasked with finding solutions to the equation 2x + 3y = 8. There are multiple solutions that satisfy this equation. Let's explore four of them:
Solution 1: Using Integer Values
Solution 2: Using Fractional Values
Solution 3: Using a Variable for y
Solution 4: Using Graphical Method
Conclusion: The equation 2x + 3y = 8 has multiple solutions, including both integer and fractional values of x and y. Additionally, solutions can also be represented using variables. Graphically, the solutions are the points where the line intersects the axes.
Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Graph of 9x – 5y + 160 = 0
To graph the equation 9x – 5y + 160 = 0, we'll first rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Step 1: Rewrite the equation in slope-intercept form
9x – 5y + 160 = 0
Subtract 9x from both sides:
-5y = -9x - 160
Divide both sides by -5 to isolate y:
y = (9/5)x + 32
Now we have the equation in slope-intercept form.
Step 2: Identify the slope and y-intercept
The slope (m) is 9/5 and the y-intercept (b) is 32.
Step 3: Plot the y-intercept and use the slope to find additional points
Now, let's plot the y-intercept at (0, 32). From there, we'll use the slope to find another point. The slope of 9/5 means that for every 5 units we move to the right along the x-axis, we move 9 units upwards along the y-axis.
So, starting from (0, 32), if we move 5 units to the right, we move 9 units up to get the next point.
Step 4: Plot the points and draw the line
Plot the y-intercept at (0, 32) and the next point at (5, 41). Then, draw a line through these points to represent the graph of the equation.
Finding the value of y when x = 5
To find the value of y when x = 5, we'll substitute x = 5 into the equation and solve for y.
9x – 5y + 160 = 0
9(5) – 5y + 160 = 0
45 – 5y + 160 = 0
Combine like terms:
-5y + 205 = 0
Subtract 205 from both sides:
-5y = -205
Divide both sides by -5 to solve for y:
y = 41
So, when x = 5, y = 41.
Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Finding Solutions of Line AB Equation
Given Information:
Procedure:
1. Identify Points on Line AB:
2. Determine Coordinates:
3. Substitute Coordinates:
4. Verify Solutions:
Example:
Conclusion:
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Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Let's denote:
The total fare can be calculated as the sum of the initial fare and the fare for the subsequent distance.
So, the equation can be expressed as:
y=10+6(x−1)y=10+6(x−1)
Where:
Now, let's substitute x=15x=15 into the equation to find the total fare for a 15 km journey.
y=10+6(15−1)y=10+6(15−1) y=10+6(14)y=10+6(14) y=10+84y=10+84 y=94y=94
The total fare for a 15 km journey would be Rs. 94.
Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Understanding Linear Equations: Linear equations are fundamental in mathematics, representing straight lines on a coordinate plane. They're expressed in the form of ax+b=0ax+b=0, where aa and bb are constants.
Identifying Axis: In the context of linear equations, the term "axis" typically refers to either the x-axis or the y-axis on a Cartesian plane.
Analyzing the Equation: The linear equation provided is x−2=0x−2=0.
Finding the Axis: To determine which axis the given linear equation is parallel to, let's analyze the equation:
Equation Form:
Solving for x:
Interpretation:
Conclusion: The linear equation x−2=0x−2=0 is parallel to the y-axis.
Answered on 18 Apr Learn Polynomials
Nazia Khanum
Factorization of Polynomials Using Factor Theorem
Introduction
Factorization of polynomials is a fundamental concept in algebra that helps in simplifying expressions and solving equations. The Factor Theorem is a powerful tool that aids in factorizing polynomials.
Factor Theorem
The Factor Theorem states that if f(c)=0f(c)=0, then (x−c)(x−c) is a factor of the polynomial f(x)f(x).
Factorization of Polynomial x3−6x2+3x+10x3−6x2+3x+10
Step 1: Find Potential Roots
Step 2: Test Roots Using Factor Theorem
Step 3: Synthetic Division
Step 4: Factorization
Factorization of x3−6x2+3x+10x3−6x2+3x+10
Potential Roots:
Testing Roots:
Synthetic Division:
Perform synthetic division:
(x3−6x2+3x+10)÷(x+2)(x3−6x2+3x+10)÷(x+2)
This yields the quotient x2−8x+5x2−8x+5.
Factorization of Quotient:
Final Factorization:
Factorization of Polynomial 2y3−5y2−19y2y3−5y2−19y
Potential Roots:
Testing Roots:
Synthetic Division:
Perform synthetic division:
(2y3−5y2−19y)÷y(2y3−5y2−19y)÷y
This yields the quotient 2y2−5y−192y2−5y−19.
Factorization of Quotient:
Final Factorization:
Conclusion
Factorizing polynomials using the Factor Theorem involves identifying potential roots, testing them, performing synthetic division, and factoring the resulting quotient. This method simplifies complex expressions and aids in solving polynomial equations effectively.
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Answered on 18 Apr Learn Polynomials
Nazia Khanum
What is the number of zeros of the quadratic equation x2+4x+2x2+4x+2?
Answer:
Quadratic Equation: x2+4x+2x2+4x+2
To determine the number of zeros of the quadratic equation, we can use the discriminant method:
Discriminant Formula:
Calculating Discriminant:
Interpreting the Discriminant:
Result:
Conclusion: The number of zeros of the quadratic equation x2+4x+2x2+4x+2 is two.
read lessAnswered on 18 Apr Learn Polynomials
Nazia Khanum
Determining the Value of k
Introduction: To find the value of k when (x – 1) is a factor of the polynomial 4x^3 + 3x^2 – 4x + k, we'll utilize the Factor Theorem.
Factor Theorem: If (x – c) is a factor of a polynomial, then substituting c into the polynomial should result in zero.
Procedure:
Step-by-Step Solution:
Substitute x=1x=1:
Solve for k:
Conclusion: The value of k when (x – 1) is a factor of the given polynomial is k=−3k=−3.
Answered on 18 Apr Learn Polynomials
Nazia Khanum
Solution: Finding Values of a and b
Given Problem: If x3+ax2–bx+10x3+ax2–bx+10 is divisible by x2–3x+2x2–3x+2, we need to find the values of aa and bb.
Solution Steps:
Step 1: Determine the factors of the divisor
Given divisor: x2–3x+2x2–3x+2
We need to find two numbers that multiply to 22 and add up to −3−3.
The factors of 22 are 11 and 22.
So, the factors that add up to −3−3 are −2−2 and −1−1.
Hence, the divisor factors are (x–2)(x–2) and (x–1)(x–1).
So, the divisor can be written as (x–2)(x–1)(x–2)(x–1).
Step 2: Use Remainder Theorem
If f(x)=x3+ax2–bx+10f(x)=x3+ax2–bx+10 is divisible by (x–2)(x–1)(x–2)(x–1), then the remainder when f(x)f(x) is divided by x2–3x+2x2–3x+2 is zero.
According to Remainder Theorem, if f(x)f(x) is divided by x2–3x+2x2–3x+2, then the remainder is given by f(2)f(2) and f(1)f(1) respectively.
Step 3: Find the value of aa
Substitute x=2x=2 into f(x)f(x) and equate it to 00 to find the value of aa.
f(2)=23+a(2)2–b(2)+10f(2)=23+a(2)2–b(2)+10
0=8+4a–2b+100=8+4a–2b+10
18=4a–2b18=4a–2b
4a–2b=184a–2b=18
Step 4: Find the value of bb
Substitute x=1x=1 into f(x)f(x) and equate it to 00 to find the value of bb.
f(1)=13+a(1)2–b(1)+10f(1)=13+a(1)2–b(1)+10
0=1+a–b+100=1+a–b+10
11=a–b11=a–b
a–b=11a–b=11
Step 5: Solve the equations
Now we have two equations:
We can solve these equations simultaneously to find the values of aa and bb.
Step 6: Solve the equations
Equation 1: 4a–2b=184a–2b=18
Divide by 2: 2a–b=92a–b=9
Equation 2: a–b=11a–b=11
Step 7: Solve the system of equations
Adding equation 2 to equation 1: (2a–b)+(a–b)=9+11(2a–b)+(a–b)=9+11
3a=203a=20
a=203a=320
Substitute a=203a=320 into equation 2: 203–b=11320–b=11
b=203–11b=320–11
b=20–333b=320–33
b=−133b=3−13
Step 8: Final values of aa and bb
a=203a=320
b=−133b=3−13
So, the values of aa and bb are a=203a=320 and b=−133b=3−13 respectively.
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Answered on 18 Apr Learn Polynomials
Nazia Khanum
Monomial and Binomial Examples with Degrees
Monomial Example (Degree: 82)
Binomial Example (Degree: 99)
Additional Notes:
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