Q no1(5).Solve the following pair of linear equations by the substitution method.

The substitution method is a way to solve a pair of linear equations by expressing one variable in terms of the other and then substituting it into the second equation. Here’s a step-by-step guide to solving a pair of linear equations using this method:
Steps for Solving by Substitution Method
Let’s consider the system of equations:
ax+by=cax + by = cax+by=c dx+ey=fdx + ey = fdx+ey=f
1. Express one variable in terms of the other
o Pick one of the equations and solve for one variable in terms of the other.
o Suppose we solve for xxx in terms of yyy: x=some expression in terms of yx = \text{some expression in terms of } yx=some expression in terms of y
2. Substitute into the second equation
o Replace xxx in the second equation with the expression obtained in Step 1.
o This will give an equation with only one variable (yyy).
3. Solve for the remaining variable
o Solve the equation obtained in Step 2 to get the value of yyy.
4. Find the value of the other variable
o Substitute the value of yyy back into the equation found in Step 1 to get xxx.
5. Write the final solution
o The values of xxx and yyy form the solution (x,y)(x, y)(x,y).
________________________________________
Example
Solve the system of equations using the substitution method:
x+2y=10x + 2y = 10x+2y=10 3x−y=53x - y = 53x−y=5
Step 1: Express one variable in terms of the other
Solve for xxx from the first equation:
x=10−2yx = 10 - 2yx=10−2y
Step 2: Substitute into the second equation
Substituting x=10−2yx = 10 - 2yx=10−2y into 3x−y=53x - y = 53x−y=5:
3(10−2y)−y=53(10 - 2y) - y = 53(10−2y)−y=5
Step 3: Solve for yyy
Expanding:
30−6y−y=530 - 6y - y = 530−6y−y=5 30−7y=530 - 7y = 530−7y=5 −7y=−25-7y = -25−7y=−25 y=257y = \frac{25}{7}y=725
Step 4: Solve for xxx
Substituting y=257y = \frac{25}{7}y=725 into x=10−2yx = 10 - 2yx=10−2y:
x=10−2×257x = 10 - 2 \times \frac{25}{7}x=10−2×725 x=10−507x = 10 - \frac{50}{7}x=10−750 x=707−507x = \frac{70}{7} - \frac{50}{7}x=770−750 x=207x = \frac{20}{7}x=720
Final Answer
(207,257)\left(\frac{20}{7}, \frac{25}{7}
ight)(720,725)
This is the solution to the given system of equations using the substitution method.
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