Linear Equations Quiz

Which of the following is a system of linear equations in two variables?

• 2x+y=5,  xy=10

• x^2+y=7,  2x+y=3

• 3x+2y=6,  4x−y=5

• x^2+y^2=9

Answer explanation

The correct choice is 3x+2y=6, 4x−y=5, as both equations are linear and involve only the variables x and y. The other options include non-linear equations or products of variables, which do not fit the criteria.

The standard form of a linear equation in two variables is:

• ax^2+by=c

• ax+by=c

• a/x+b/y=c

• ax+by+c=0

Answer explanation

The standard form of a linear equation in two variables is represented as ax + by = c, where a and b are coefficients, and c is a constant. This format clearly shows the relationship between the variables x and y.

If the lines represented by 2x+3y=5 and 4x+6y=10 are compared, they are:

• Coincident

• Parallel

• Intersecting at one point

• Perpendicular

Answer explanation

The equations 2x + 3y = 5 and 4x + 6y = 10 represent the same line, as the second equation is a multiple of the first. Therefore, the lines are coincident, meaning they overlap completely.

A system of linear equations has no solution if the lines are:

• Parallel

• Coincident

• Intersecting

• Overlapping partially

Answer explanation

A system of linear equations has no solution when the lines are parallel, as they never intersect. Coincident lines have infinite solutions, intersecting lines have one solution, and overlapping partially also implies a solution exists.

A system of linear equations has infinite solutions if the lines are:

• Parallel

• Coincident

• Intersecting

• Perpendicular

Answer explanation

A system of linear equations has infinite solutions when the lines are coincident, meaning they lie on top of each other. Parallel lines have no solutions, while intersecting lines have exactly one solution.

If two lines intersect at a single point, the system is:

• Consistent, independent

• Consistent, dependent

• Inconsistent

• None

Answer explanation

When two lines intersect at a single point, they have exactly one solution, making the system consistent. Since they are not parallel and do not coincide, they are also independent. Thus, the correct answer is 'Consistent, independent'.

Which method involves multiplying one or both equations to eliminate a variable?

• Substitution method

• Graphical method

• Elimination method

• Matrix method

Answer explanation

The elimination method involves manipulating equations, often by multiplying them, to eliminate one variable, making it easier to solve for the other variable. This distinguishes it from the substitution and graphical methods.

The substitution method involves:

• Multiplying equations

• Expressing one variable in terms of the other

• Drawing graphs

• Using determinants

Answer explanation

The substitution method involves expressing one variable in terms of the other, allowing you to solve equations systematically. This is the key step in the substitution process, making it the correct choice.

If the determinant of the coefficient matrix is zero, the system is:

• Always consistent

• Inconsistent or dependent

• Always independent

• None

Answer explanation

If the determinant of the coefficient matrix is zero, the system may have no solutions (inconsistent) or infinitely many solutions (dependent). Thus, the correct choice is 'Inconsistent or dependent'.

Which of the following is not a method to solve linear equations?

• Elimination

• Substitution

• Cross multiplication

• Differentiation

Answer explanation

Differentiation is a calculus method used for finding rates of change, not for solving linear equations. The other methods, elimination and substitution, are standard techniques for solving such equations.