Linear Equations With One And Two Unkno
Course Notes Linear Math & Matrices
PSB – Dr. H. Schellinx
Linear equations
As we have seen, a linear equation with n different variables, say x1, x2 , x3,..., xn , can always be written in the equivalent standard form a1 x1 + a2 x2 + a3 x3 +... + an xn = c , where c is a constant, the xi are the unknowns and the ci are coefficients.
Here are some examples: 5x − 3y = z + w − 6 is a linear equation with 4 unknowns (x,y,w and z). It has the equivalent standard form 5x − 3y − w − z = −6 .
−3y = y + 2x − 7 is a linear equation with 2 unknowns (x and y). It has the equivalent standard form 2x + 4y = 7 .
Linear equations with one unknown
Linear equations with only one variable are the simplest ones. We can always write such an equation in the standard form ax = b . Here x is the only variable (the unknown). The coefficient a and the constant b are both real numbers.
1. If a ≠ 0 , the equation has precisely one solution: x =
b
.
a
2. If a = 0 and b ≠ 0 , the equation has no solution. It is a contradiction. 3. If a = 0 and b = 0 , all real numbers satisfy the equation. It is an identity.
Recall that the root or zero of a function f(x) is a member x of the domain of f such that f(x) = 0 (we say that f(x) vanishes at x). The solution of the linear equation with one unknown ax = b , corresponds to the
PSB – Dr. H. Schellinx
Linear equations
As we have seen, a linear equation with n different variables, say x1, x2 , x3,..., xn , can always be written in the equivalent standard form a1 x1 + a2 x2 + a3 x3 +... + an xn = c , where c is a constant, the xi are the unknowns and the ci are coefficients.
Here are some examples: 5x − 3y = z + w − 6 is a linear equation with 4 unknowns (x,y,w and z). It has the equivalent standard form 5x − 3y − w − z = −6 .
−3y = y + 2x − 7 is a linear equation with 2 unknowns (x and y). It has the equivalent standard form 2x + 4y = 7 .
Linear equations with one unknown
Linear equations with only one variable are the simplest ones. We can always write such an equation in the standard form ax = b . Here x is the only variable (the unknown). The coefficient a and the constant b are both real numbers.
1. If a ≠ 0 , the equation has precisely one solution: x =
b
.
a
2. If a = 0 and b ≠ 0 , the equation has no solution. It is a contradiction. 3. If a = 0 and b = 0 , all real numbers satisfy the equation. It is an identity.
Recall that the root or zero of a function f(x) is a member x of the domain of f such that f(x) = 0 (we say that f(x) vanishes at x). The solution of the linear equation with one unknown ax = b , corresponds to the