Solve the system Ax=b, where A=\left[\begin{array}{ccc} 3&4\ {5}&{6}\end{array} ight] and b=\left[\begin{array}{ccc} 4.7\ {7}\end{array} ight]. What value is the first entry of x?

Question

Solve the system Ax=b, where A=\left[\begin{array}{ccc} 3&4\ {5}&{6}\end{array}ight] and b=\left[\begin{array}{ccc} 4.7\ {7}\end{array}ight]. What value is the first entry of x?

BlackTom AI · Accepted Answer

Let me restate the problem in my own words: we need to solve the linear system A x = b where A = [[3,4],[5,6]] and b = [4.7, 7], and determine the first entry of x.

A strategy is to compute the solution x = A^{-1} b if A is invertible. First, check the determinant of A to ensure invertibility: det(A) = 3*6 - 4*5 = 18 - 20 = -2, which is nonzero, so A is invertible.

Next, find the inverse A^{-1}. For a 2x2 matrix [[a,b],[c,d]], the inverse is (1/det) * [[d,-b],[-c,a]]. Applying this to A = [[3,4],[5,6]] with det = -2 gives
A^{-1} = (1/(-2)) * [[6, -4], [-5, 3]] = [[-3, 2], [2.5, -1.5]].

Now compute x = A^{-1} b. With b = [4.7, 7], multiply:
- x1 = (-3)*4.7 + 2*7 = -14.1 + 14 = -0.1.
- x2 = (2.5)*4.7 + (-1.5)*7 = 11.75 - 10.5 = 1.25.

Therefore, the first entry of x is -0.1. If you’d like a quick cross-check, you can verify that A times x yields b: A * x ≈ [[3,4],[5,6]] * [-0.1, 1.25]ᵗ ≈ [4.7, 7]ᵗ, confirming consistency.

ETF2700 - ETF5970 - S2 2025 Quiz 2

Solve the system Ax=b, where A=\left[\begin{array}{ccc} 3&4\\ {5}&{6}\end{array}\right] and b=\left[\begin{array}{ccc} 4.7\\ {7}\end{array}\right]. What value is the first entry of x?

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