A system of equations with variables x and y is given: 3y + bx = 4 4x-y = 8 at what value of b will the system have no solutions?
If a system of equations with unknowns x and y is given, then 3 cases are possible: 1) there is one (unique) solution; 2) there are infinitely many solutions; 3) there is no solution. We are interested in case 3), which is realized when the coefficients of one equation are obtained by dividing or multiplying the corresponding coefficients of another equation, and free numbers do not have this property.
In setting the coefficients of the first equation b * x + 3 * y = 4 are b and 3, and the free term is 4. Similarly, the coefficients of the second equation 4 * x – y = 8 are 4 and – 1, and the free term is 8. Clearly , that in the first equation the coefficient 3 at y is obtained by multiplying the corresponding coefficient –1 by –3. Then in the first equation the coefficient b at x should also be obtained by multiplying the corresponding coefficient 4 by –3, that is, the equality b = 4 * (–3) should be fulfilled, that is, b = –12.
Now let’s check if the free term 4 of the first equation is obtained by multiplying the free term 8 of the second equation by –3. We have 8 * (–3) = –24 ≠ 4.
Thus, for b = –12, the conditions of case 3) are fully satisfied.
Answer: For b = –12 the system will have no solutions.
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