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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
To draw the graph of the equation 2x−3y=122x−3y=12, let's first rewrite it in slope-intercept form, which is y=mx+by=mx+b, where mm is the slope and bb is the y-intercept.
2x−3y=122x−3y=12
−3y=−2x+12−3y=−2x+12
y=23x−4y=32x−4
Y-intercept: When x=0x=0,
y=23(0)−4y=32(0)−4
y=−4y=−4
So, the y-intercept is (0, -4).
Slope: The coefficient of xx is 2332, which represents the slope.
For every increase of 1 in xx, yy increases by 2332.
For every decrease of 1 in xx, yy decreases by 2332.
Now, let's plot some points to draw the graph:
x = 3: y=23(3)−4=2−4=−2y=32(3)−4=2−4=−2
Point: (3, -2)
x = 6: y=23(6)−4=4−4=0y=32(6)−4=4−4=0
Point: (6, 0)
x = -3: y=23(−3)−4=−2−4=−6y=32(−3)−4=−2−4=−6
Point: (-3, -6)
With these points, we can draw a straight line passing through them.
To find where the graph intersects the x-axis, we set y=0y=0 and solve for xx:
0=23x−40=32x−4
23x=432x=4
x=4×32x=24×3
x=6x=6
So, the graph intersects the x-axis at x=6x=6, which corresponds to the point (6, 0).
To find where the graph intersects the y-axis, we set x=0x=0 and solve for yy:
y=23(0)−4y=32(0)−4
y=−4y=−4
So, the graph intersects the y-axis at y=−4y=−4, which corresponds to the point (0, -4).
This information helps us visualize and understand the behavior of the equation 2x−3y=122x−3y=12 on the coordinate plane.
Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Problem Analysis:
Given the equation 2x−y=p2x−y=p and a solution point (1,−2)(1,−2), we need to find the value of pp.
Solution:
Step 1: Substitute the Given Solution into the Equation
Substitute the coordinates of the given solution point (1,−2)(1,−2) into the equation:
2(1)−(−2)=p2(1)−(−2)=p
Step 2: Solve for pp
2+2=p2+2=p 4=p4=p
Step 3: Final Result
p=4p=4
Conclusion:
The value of pp for the equation 2x−y=p2x−y=p when the point (1,−2)(1,−2) is a solution is 44.
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Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Graph of the Equation x - y = 4
Graphing the Equation:
To draw the graph of the equation x−y=4x−y=4, we'll first rewrite it in slope-intercept form, which is y=mx+by=mx+b, where mm is the slope and bb is the y-intercept.
Given equation: x−y=4x−y=4
Rewriting in slope-intercept form:
y=x−4y=x−4
Now, let's plot the graph using this equation.
Plotting the Graph:
Find y-intercept:
Set x=0x=0 in the equation y=x−4y=x−4
y=0−4y=0−4
y=−4y=−4
So, the y-intercept is at the point (0,−4)(0,−4).
Find x-intercept:
To find the x-intercept, set y=0y=0 in the equation y=x−4y=x−4.
0=x−40=x−4
x=4x=4
So, the x-intercept is at the point (4,0)(4,0).
Drawing the Graph:
Now, plot the points (0,−4)(0,−4) and (4,0)(4,0) on the Cartesian plane and draw a straight line passing through these points. This line represents the graph of the equation x−y=4x−y=4.
Intersecting with the x-axis:
To find where the graph line meets the x-axis, we need to find the point where y=0y=0.
Substitute y=0y=0 into the equation x−y=4x−y=4:
x−0=4x−0=4
x=4x=4
So, when the graph line meets the x-axis, the coordinates of the point are (4,0)(4,0).
Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Graphing the Equation x + 2y = 6
To graph the equation x+2y=6x+2y=6, we'll first rewrite it in slope-intercept form (y=mx+by=mx+b):
x+2y=6x+2y=6 2y=−x+62y=−x+6 y=−12x+3y=−21x+3
Plotting the Graph
To plot the graph, we'll identify two points and draw a line through them:
Intercept Method:
Slope Method: From the slope-intercept form y=−12x+3y=−21x+3, the slope is -1/2, meaning the line decreases by 1 unit in the y-direction for every 2 units in the x-direction.
Plotting the Points and Drawing the Line
Using the intercepts and the slope, we plot the points (0, 3) and (-6, 0), then draw a line through them.
Finding the Value of x when y = -3
Given y=−3y=−3, we substitute this value into the equation y=−12x+3y=−21x+3 and solve for x:
−3=−12x+3−3=−21x+3 −12x=−3−3−21x=−3−3 −12x=−6−21x=−6 x=−6×(−2)x=−6×(−2) x=12x=12
Conclusion
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.
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Answered on 18 Apr Learn Polynomials
Nazia Khanum
Problem Statement: Find the value of x2+y2x2+y2, given x+y=12x+y=12 and xy=32xy=32.
Solution:
Step 1: Understanding the problem
Step 2: Solving the equations
Step 3: Finding the values of xx and yy
Step 4: Finding corresponding values of yy
Step 5: Calculating x2+y2x2+y2
Step 6: Presenting the solution
Final Answer:
This structured approach helps in solving the problem systematically, ensuring accuracy and clarity.
Answered on 18 Apr Learn Polynomials
Nazia Khanum
Problem Analysis: Given equations:
We need to find the value of 9x2+4y29x2+4y2.
Solution:
Step 1: Find the values of xx and yy
To solve the system of equations, we can use substitution or elimination method.
From equation (2), xy=6xy=6, we can express yy in terms of xx: y=6xy=x6
Substitute this expression for yy into equation (1): 3x+2(6x)=123x+2(x6)=12
Now solve for xx:
3x+12x=123x+x12=12 3x2+12=12x3x2+12=12x 3x2−12x+12=03x2−12x+12=0
Divide the equation by 3: x2−4x+4=0x2−4x+4=0
Factorize: (x−2)2=0(x−2)2=0
So, x=2x=2.
Now, substitute x=2x=2 into equation (2) to find yy: 2y=62y=6 y=3y=3
So, x=2x=2 and y=3y=3.
Step 2: Find the value of 9x2+4y29x2+4y2
Substitute the values of xx and yy into the expression 9x2+4y29x2+4y2: 9(2)2+4(3)29(2)2+4(3)2 9(4)+4(9)9(4)+4(9) 36+3636+36 7272
Conclusion: The value of 9x2+4y29x2+4y2 is 7272.
Answered on 18 Apr Learn Polynomials
Nazia Khanum
Perimeter Calculation for Rectangle with Given Area
Given Information:
Step 1: Determine the Dimensions
To calculate the perimeter of a rectangle, we need to know its length and width. We can find these dimensions using the area provided.
Step 2: Factorize the Area
Factorize the given quadratic expression 25x2−35x+1225x2−35x+12 to find its factors, which represent the possible lengths and widths of the rectangle.
Step 3: Use Factorization to Find Dimensions
Once the quadratic expression is factorized, identify the pairs of factors that, when multiplied, give the area of the rectangle. These pairs represent possible lengths and widths.
Step 4: Calculate Perimeter
With the length and width of the rectangle known, calculate the perimeter using the formula:
Perimeter=2×(Length+Width)Perimeter=2×(Length+Width)
Step 5: Finalize
Plug in the values of length and width into the perimeter formula to obtain the final result.
Let's proceed with these steps to find the perimeter.
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Answered on 18 Apr Learn Polynomials
Nazia Khanum
Given:
To Find:
Approach:
Step-by-Step Solution:
Find x3+y3+z3x3+y3+z3:
Find (x+y)(y+z)(z+x)(x+y)(y+z)(z+x):
Substitute values into the expression:
Final Answer:
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