A nxn homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.

A **system** of linear equations ax + by + c = 0 and dx + ey + g = 0 will **have a unique solution** if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point. i.e., if the two lines are neither parallel nor coincident. Essentially, the slopes of the two lines **should** be different.

Likewise, what is the condition for no solution? If there are infinitely many **solutions** of the given pair of linear equations, the equations are called dependent (consistent). If the lines are parallel, there is **no solution** for the pair of linear equations. If there is **no solution** of the given pair of linear equations, the equations are called inconsistent.

Herein, what is a unique solution?

A system has a **unique solution** when it is consistent and the number of variables is equal to the number of nonzero rows. If the rref of the matrix for the system is , the **solution** is the single point ( 2, 1, 3 ) or x=2, y=1, z=3.

What does infinitely many solutions look like?

The first **is** when we have **what is** called **infinite solutions**. This happens when all numbers **are solutions**. This situation means that there **is** no one **solution**. The equation 2x + 3 = x + x + 3 **is** an example of an equation that has an **infinite** number of **solutions**.

### What does it mean when determinant is zero?

If the determinant of a square matrix n×n A is zero, then A is not invertible. [When the determinant of a matrix is nonzero, the linear system it represents is linearly independent.] When the determinant of a matrix is zero, its rows are linearly dependent vectors, and its columns are linearly dependent vectors.

### How do you solve system of equations?

Here’s how it goes: Step 1: Solve one of the equations for one of the variables. Let’s solve the first equation for y: Step 2: Substitute that equation into the other equation, and solve for x. Step 3: Substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.

### What is no solution?

The solution x = 0 means that the value 0 satisfies the equation, so there is a solution. “No solution” means that there is no value, not even 0, which would satisfy the equation. This is because there is truly no solution—there are no values for x that will make the equation 12 + 2x – 8 = 7x + 5 – 5x true.

### What is Cramer’s rule matrices?

Cramer’s Rule for a 2×2 System (with Two Variables) Cramer’s Rule is another method that can solve systems of linear equations using determinants. In terms of notations, a matrix is an array of numbers enclosed by square brackets while determinant is an array of numbers enclosed by two vertical bars.

### What makes a matrix consistent?

In mathematics and in particularly in algebra, a linear or nonlinear system of equations is called as consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, that when substituted into each of the equations makes each equation hold true as an identity.

### How many solutions does the system have?

Systems of linear equations can only have 0, 1, or an infinite number of solutions. These two lines cannot intersect twice. The correct answer is that the system has one solution.

### What is a system of equations that has no solution?

Since parallel lines never cross, then there can be no intersection; that is, for a system of equations that graphs as parallel lines, there can be no solution. This is called an “inconsistent” system of equations, and it has no solution.

### What is the condition for infinitely many solutions?

An equation can have infinitely many solutions when it should satisfy some conditions. The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept. If the two lines have the same y-intercept and the slope, they are actually the same exact line.