(a) The sum of the digits of a 2-digit number is 12. Seven times the number is equal to four times the number obtained by reversing the order of the digits. Find the number.
OR
(b) Find the values of x and y from the following pair of linear equations :
2x+y=0.8 x+y7=10
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Step-by-Step Solution
Step 1: Set up equations for the 2-digit number problem
Let the two-digit number be represented as 10t + u, where t is the tens digit and u is the units digit. The first condition states that the sum of the digits is 12, which gives us the equation t+u=12. The second condition states that seven times the number is equal to four times the number obtained by reversing the digits. Reversing the digits gives the number 10u + t. This leads to the equation 7(10t+u)=4(10u+t).
Step 2: Simplify the second equation
Expand both sides of the second equation: 70t+7u=40u+4t. Rearrange the terms to group t and u on opposite sides: 70t−4t=40u−7u, which simplifies to 66t=33u. Dividing both sides by 33 gives us a simpler relationship between t and u: 2t=u.
Step 3: Solve the system of equations
Now we have a system of two linear equations: t+u=12 and u=2t. Substitute the expression for u from the second equation into the first equation: t+(2t)=12. This simplifies to 3t=12, so t=4. Substitute the value of t back into u=2t to find u=2(4)=8.
Step 4: Form the number
With t=4 and u=8, the original two-digit number is 10t+u=10(4)+8=40+8=48.