Free Algebra
Arithmetic Operations with Numerical Fractions
Multiplying a Polynomial by a Monomial
Solving Linear Equation
Solving Linear Equations
Solving Inequalities
Solving Compound Inequalities
Solving Systems of Equations Using Substitution
Simplifying Fractions 3
Factoring quadratics
Special Products
Writing Fractions as Percents
Using Patterns to Multiply Two Binomials
Adding and Subtracting Fractions
Solving Linear Inequalities
Adding Fractions
Solving Systems of Equations -
Exponential Functions
Integer Exponents
Example 6
Dividing Monomials
Multiplication can Increase or Decrease a Number
Graphing Horizontal Lines
Simplification of Expressions Containing only Monomials
Decimal Numbers
Negative Numbers
Subtracting Polynomials
Adding and Subtracting Fractions
Powers of i
Multiplying and Dividing Fractions
Simplifying Complex Fractions
Finding the Coordinates of a Point
Fractions and Decimals
Rational Expressions
Solving Equations by Factoring
Slope of a Line
Percent Introduced
Reducing Rational Expressions to Lowest Terms
The Hyperbola
Standard Form for the Equation of a Line
Multiplication by 75
Solving Quadratic Equations Using the Quadratic Formula
Raising a Product to a Power
Solving Equations with Log Terms on Each Side
Monomial Factors
Solving Inequalities with Fractions and Parentheses
Division Property of Square and Cube Roots
Multiplying Two Numbers Close to but less than 100
Solving Absolute Value Inequalities
Equations of Circles
Percents and Decimals
Integral Exponents
Linear Equations - Positive and Negative Slopes
Multiplying Radicals
Factoring Special Quadratic Polynomials
Simplifying Rational Expressions
Adding and Subtracting Unlike Fractions
Graphuing Linear Inequalities
Linear Functions
Solving Quadratic Equations by Using the Quadratic Formula
Adding and Subtracting Polynomials
Adding and Subtracting Functions
Basic Algebraic Operations and Simplification
Simplifying Complex Fractions
Axis of Symmetry and Vertices
Factoring Polynomials with Four Terms
Evaluation of Simple Formulas
Graphing Systems of Equations
Scientific Notation
Lines and Equations
Horizontal and Vertical Lines
Solving Equations by Factoring
Solving Systems of Linear Inequalities
Adding and Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Fractions
Solving Linear Equations
Simple Trinomials as Products of Binomials
Solving Nonlinear Equations by Factoring
Solving System of Equations
Exponential Functions
Computing the Area of Circles
The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
Squaring a Difference
Changing the Sign of an Exponent
Adding Fractions
Powers of Radical Expressions
Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Solving System of Equations - Two Lines

Solve a System of Equations: Elimination and Substitution:

Two points determine a line. – Two lines determine a point.

Two points determine a line.

A single line can be drawn through 2 points .

Two lines determine a point.

Graphically an infinite number of lines can be drawn through one point which form a “family of lines” The vertical line (x = h) and the horizontal line (y = k) form the base-lines for this “family”

A “system of equations” is defined by the set of all of the lines that intersect at the point (h, k) and the solution which can be determined by the equations of two different lines. The simplest of these are { x = h, y = k } which is called the “solution set” of the system of equations. The solution is also defined by the point of intersection: { (h, k) }.

In any system of equations (family of lines) the following are true statements.

1. A linear equation may be multiplied by a real number and form another linear equation through the same point. (a coincident line).

2. Two intersecting linear equations may be added to form the equation of another line through the same point on a graph.

3. One equation can be “solved for one of the variables” and that expression can then be substituted in another equation in place of the variable.

 NOTE: [In problems use only for variables with coefficient of ±1.]

Elimination (Addition) Method:

Use the same basic properties that you used with equations of one variable to solve a system of equations (find the solution point). In the this method add the equations, or multiply then add, to eliminate one variable to solve for the other. Replace that value to complete the solution.

Example 1:

Add the two equations: multiply:

Replace x = 3 in equation 1: (3) + y = 8 or y = 5

Always check in the other equation: (3) − (5) = −2   The solution point: { ( 3, 5) }

Example 2:

Add the two equations: multiply recip:

Replace x = 2 in equation 2: (2) − y = 1 or

Always check in the other equation: 2(3) − (5) = 1

The solution is the point (2, 1): or S = { ( 2, 1) }

Example 3:

Multiple the second equation by 3 obtain:

Add the equations gives

Replace x = 2 in equation 2: 3 (2) − y = 1 or

Always check by replacing both in first equation: 2(2) + 3 (5) = 19

Thus, the solution is the point (2, 5): or S = { ( 2, 5) }

Example 4:

When both equations have coefficients that don’t seem readily compatible we must multiply each equation so that one of the variables has coefficients that are “equal and opposite”.

Multiply: and find

Now, add and get. Then

Replace x = 3 in the second equation: 2(3) − 3y = − 9 finds

Always check by replacing both in first equation: 3(3) + 2(5) = 19 The solution point: { (3, 5) }

All Right Reserved. Copyright 2005-2023