Solve these pairs of simultaneous equation both analytically and graphically 5x+2y=1,3x+5y=13 other questions y=x^2_3x+4, y=3x_2

To tackle the pairs of simultaneous equations you've provided, we'll solve them both analytically and graphically. Let's start with the first pair: 5x + 2y = 1 and 3x + 5y = 13.

Analytical Solution

To solve these equations analytically, we can use the method of substitution or elimination. Here, we'll use the elimination method for clarity.

Step 1: Align the equations

We have:

• Equation 1: 5x + 2y = 1
• Equation 2: 3x + 5y = 13

Step 2: Make coefficients of y the same

To eliminate y, we can multiply Equation 1 by 5 and Equation 2 by 2:

• 5(5x + 2y) = 5(1) → 25x + 10y = 5
• 2(3x + 5y) = 2(13) → 6x + 10y = 26

Step 3: Subtract the equations

Now, we subtract the first modified equation from the second:

(6x + 10y) - (25x + 10y) = 26 - 5

This simplifies to:

-19x = 21

So, x = -21/19.

Step 4: Substitute x back to find y

Now, substitute x back into one of the original equations. Let's use Equation 1:

5(-21/19) + 2y = 1

This simplifies to:

-105/19 + 2y = 1

2y = 1 + 105/19 = 19/19 + 105/19 = 124/19

Thus, y = 62/19.

Graphical Representation

To graph these equations, we need to convert them into slope-intercept form (y = mx + b).

Equation 1: 5x + 2y = 1

Rearranging gives:

2y = -5x + 1 → y = -5/2x + 1/2.

Equation 2: 3x + 5y = 13

Rearranging gives:

5y = -3x + 13 → y = -3/5x + 13/5.

Now, you can plot these lines on a graph. The point where they intersect is the solution to the simultaneous equations. You should find that the intersection point corresponds to the analytical solution we calculated: (-21/19, 62/19).

Next Set of Equations

Now, let's look at the second set of equations: y = x^2 - 3x + 4 and y = 3x - 2.

Analytical Approach

To find the points of intersection, we set the equations equal to each other:

x^2 - 3x + 4 = 3x - 2.

Step 1: Rearranging the equation

Bringing all terms to one side gives:

x^2 - 6x + 6 = 0.

Step 2: Solve using the quadratic formula

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. Here, a = 1, b = -6, and c = 6:

x = (6 ± √((-6)² - 4(1)(6))) / 2(1) = (6 ± √(36 - 24)) / 2 = (6 ± √12) / 2 = (6 ± 2√3) / 2 = 3 ± √3.

Step 3: Find corresponding y values

Now, substitute x back into either equation to find y. Using y = 3x - 2:

• For x = 3 + √3: y = 3(3 + √3) - 2 = 9 + 3√3 - 2 = 7 + 3√3.
• For x = 3 - √3: y = 3(3 - √3) - 2 = 9 - 3√3 - 2 = 7 - 3√3.

Graphical Interpretation

To graph these equations, plot the parabola represented by y = x^2 - 3x + 4 and the line y = 3x - 2. The points where they intersect will be the solutions we found analytically: (3 + √3, 7 + 3√3) and (3 - √3, 7 - 3√3).

In summary, we approached both pairs of simultaneous equations through analytical methods and graphical representation, providing a comprehensive understanding of how to solve them effectively. If you have any further questions or need clarification on any step, feel free to ask!