Algebra (Comprehensive Review)

Chapter 1

Algebra

Course: Sat Prep
Subject: Math
Chapter: Algebra

By the end of this lesson, you will be able to:

• Define and solve systems of linear equations.
• Solve systems of equations using the graphing method.
• Solve systems using the substitution method.
• Solve systems using the elimination (addition) method.
• Identify systems with one solution, no solution, or infinitely many solutions.
• Solve real-world word problems involving systems of equations.
• Interpret the meaning of solutions in the context of a problem.

What Is a System of Linear Equations?

A system of linear equations consists of two or more linear equations that share the same variables. The objective is to find the value of each variable that satisfies all equations in the system simultaneously.

A solution to a system is called the point of intersection because it is the point where the graphs of the equations meet.

The general form of a system of two linear equations is

$\begin{cases} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2 \end{cases}$

where

• $a_1, a_2, b_1,$ and $b_2$ are constants,
• $x$ and $y$ are variables,
• $a_1$ and $a_2$ are not both zero.

A system may have:

• Exactly one solution
• No solution
• Infinitely many solutions

Example

Consider the system

$\begin{cases} x+y=7\\ x-y=1 \end{cases}$

The solution is

$(x,y)=(4,3)$

because

$4+3=7$

and

$4-3=1.$

Since the ordered pair satisfies both equations, it is the solution to the system.

Types of Solutions

Every system of linear equations falls into one of three categories.

| Type of System | Number of Solutions | Graph | | Consistent Independent | One | Two lines intersect once | | Inconsistent | None | Parallel lines | | Consistent Dependent | Infinitely many | Same line |

Understanding these categories helps determine whether a system has a unique solution before performing calculations.

Graphing Method

The graphing method involves drawing the graph of each equation on the same coordinate plane. The point where the two lines intersect represents the solution to the system.

Although graphing provides a visual understanding of systems, it is usually less accurate than algebraic methods unless the intersection occurs at integer coordinates.

Steps for the Graphing Method

• Rewrite each equation in slope-intercept form if necessary.

$y=mx+b$

• Graph both lines on the same coordinate plane.
• Identify the point where the lines intersect.
• Verify the solution by substituting the ordered pair into both equations.

Example 1

Solve the system by graphing.

$\begin{cases} y=x+1\\ y=-x+5 \end{cases}$

Substitute the two equations since both equal $y$.

$x+1=-x+5$

Add $x$ to both sides.

$2x=4$ $x=2$

Find $y$.

$y=2+1=3$

Therefore,

$\boxed{(2,3)}$

is the solution.

A graph of the two lines would show them intersecting at the point $(2,3)$.

Example 2

Solve by graphing.

$\begin{cases} y=2x-1\\ y=x+2 \end{cases}$

Set the equations equal.

$2x-1=x+2$

Subtract $x$.

$x-1=2$

Add 1.

$x=3$

Substitute into either equation.

$y=3+2=5$

The solution is

$\boxed{(3,5)}$

The graph would show the two lines intersecting at $(3,5)$.

Solving Systems Using Substitution

The substitution method is most effective when one equation is already solved for one variable or can easily be rewritten in that form.

The idea is to substitute an expression from one equation into the other equation, reducing the system to a single equation with one variable.

Steps for the Substitution Method

• Solve one equation for one variable.
• Substitute that expression into the other equation.
• Solve for the remaining variable.
• Substitute back to find the second variable.
• Verify the solution.

Example 1

Solve

$\begin{cases} y=2x+1\\ 3x+y=13 \end{cases}$

Substitute $2x+1$ for $y$.

$3x+(2x+1)=13$

Combine like terms.

$5x+1=13$

Subtract 1.

$5x=12$

Divide by 5.

$x=\frac{12}{5}$

Find $y$.

$y=2\left(\frac{12}{5}\right)+1$ $y=\frac{24}{5}+\frac{5}{5}$ $y=\frac{29}{5}$

The solution is

$\boxed{\left(\frac{12}{5},\frac{29}{5}\right)}$

Example 2

Solve

$\begin{cases} x=y+4\\ 2x+y=13 \end{cases}$

Substitute $y+4$ for $x$.

$2(y+4)+y=13$

Distribute.

$2y+8+y=13$ $3y=5$ $y=\frac53$

Find $x$.

$x=\frac53+4=\frac{17}{3}$

The solution is

$\boxed{\left(\frac{17}{3},\frac53\right)}$

Solving Systems Using Elimination

The elimination method, also called the addition method, eliminates one variable by adding or subtracting the equations.

This method is particularly useful when the coefficients of one variable are already opposites or can easily be made opposites.

Steps for the Elimination Method

• Arrange both equations in standard form.

$Ax+By=C$

• Multiply one or both equations if necessary to create opposite coefficients.
• Add or subtract the equations.
• Solve for the remaining variable.
• Substitute back to find the other variable.

Example 1

Solve

$\begin{cases} 2x+y=11\\ 2x-y=5 \end{cases}$

Add the equations.

$4x=16$ $x=4$

Substitute into the first equation.

$2(4)+y=11$ $8+y=11$ $y=3$

Therefore,

$\boxed{(4,3)}$

Example 2

Solve

$\begin{cases} 3x+2y=18\\ 3x-2y=6 \end{cases}$

Add the equations.

$6x=24$ $x=4$

Substitute into the first equation.

$12+2y=18$ $2y=6$ $y=3$

The solution is

$\boxed{(4,3)}$

Systems with No Solution

A system has no solution when the lines are parallel.

Parallel lines have

• the same slope,
• different y-intercepts.

Because parallel lines never intersect, there is no ordered pair that satisfies both equations.

Example

$\begin{cases} y=2x+3\\ y=2x-4 \end{cases}$

Both equations have slope 2.

Their y-intercepts are different.

The lines never intersect.

Therefore,

$\boxed{\text{No Solution}}$

Systems with Infinite Solutions

A system has infinitely many solutions when both equations represent the same line.

One equation is simply a multiple of the other.

Example

$\begin{cases} 2x+4y=12\\ x+2y=6 \end{cases}$

Multiply the second equation by 2.

$2x+4y=12$

Both equations are identical.

Every point on the line satisfies both equations.

Therefore,

$\boxed{\text{Infinitely Many Solutions}}$

Word Problems

Many SAT questions require translating real-life situations into systems of equations.

Example 1

A school sold 180 tickets for a play. Adult tickets cost $10 and student tickets cost$6. The total revenue was $1,320.

Let

• $a$ = number of adult tickets
• $s$ = number of student tickets

The system is

$\begin{cases} a+s=180\\ 10a+6s=1320 \end{cases}$

Solve using elimination.

Multiply the first equation by 6.

$6a+6s=1080$

Subtract.

$4a=240$ $a=60$

Substitute.

$60+s=180$ $s=120$

The school sold 60 adult tickets and 120 student tickets.

Example 2

Two numbers have a sum of 25. Their difference is 7.

Let the numbers be $x$ and $y$.

The system is

$\begin{cases} x+y=25\\ x-y=7 \end{cases}$

Add the equations.

$2x=32$ $x=16$

Substitute.

$16+y=25$ $y=9$

The two numbers are

$\boxed{16\text{ and }9}$

Choosing the Best Method

| Situation | Best Method | | Need a visual solution | Graphing | | One equation is already solved for a variable | Substitution | | Coefficients are opposites or easy to make opposites | Elimination | | Word problems | Usually Elimination or Substitution |

Choosing the appropriate method can simplify calculations and reduce the chance of errors.

Common Mistakes

Students often make the following mistakes when solving systems of equations:

• Graphing inaccurately and identifying the wrong point of intersection.
• Forgetting to substitute the first variable back into an original equation.
• Making arithmetic errors during elimination.
• Failing to recognize parallel lines as having no solution.
• Confusing identical equations with parallel equations.

Always verify your solution by substituting the ordered pair into both original equations.

Solutions

• Rewrite equations into slope-intercept form before graphing.
• If one equation is already solved for a variable, substitution is usually the fastest method.
• Elimination is often the most efficient method when coefficients are easy to eliminate.
• If variables disappear and the result is a false statement such as $5=9$, the system has no solution.
• If variables disappear and the result is a true statement such as $4=4$, the system has infinitely many solutions.
• Check your final answer in both equations to avoid careless mistakes.

Lesson Summary

A system of linear equations consists of two or more equations that share the same variables. The solution is the ordered pair that satisfies every equation in the system. Systems can be solved using graphing, substitution, or elimination, with each method being appropriate in different situations. Some systems have one unique solution, while others have no solution because the lines are parallel or infinitely many solutions because the equations represent the same line.