In this review lesson you’ll revisit the major ideas from Algebra Basics, practice mixed problems across topics, and apply them to real-life situations to solidify your skills before moving to the next section.
Algebra Basics builds on Pre-Algebra by focusing on linear relationships and early polynomials. Key skills include:
Solving multi-step equations and inequalities (including variables on both sides)
Rearranging literal equations and formulas
Understanding functions, slope, and linear forms (slope-intercept, point-slope, standard)
Graphing lines and solving systems (graphing, substitution, elimination)
Working with polynomials (adding, subtracting, basic factoring)
Tips for mixed review:
Identify the main concept first (equation, system, factoring, etc.)
Show all steps, especially when solving systems or factoring
Check solutions by substituting back or estimating
Use real-world context to verify answers make sense
1. Solve the system:
{ 2 x + 3 y = 12 x − y = 1 \begin{cases} 2x + 3y = 12 \\ x - y = 1 \end{cases} { 2 x + 3 y = 12 x − y = 1
Use substitution. Solve the second equation for x:
x = y + 1 x = y + 1 x = y + 1
Substitute into the first equation:
2 ( y + 1 ) + 3 y = 12 2 y + 2 + 3 y = 12 5 y = 10 y = 2 \begin{aligned}
2(y + 1) + 3y &= 12 \\[1em]
2y + 2 + 3y &= 12 \\[1em]
5y &= 10 \\[1em]
y &= 2
\end{aligned} 2 ( y + 1 ) + 3 y 2 y + 2 + 3 y 5 y y = 12 = 12 = 10 = 2
Back-substitute to find x:
x = 2 + 1 = 3 x = 2 + 1 = 3 x = 2 + 1 = 3
Solution: ( 3 , 2 ) (3, 2) ( 3 , 2 )
2. Factor x² + 6x + 8
Find two numbers that multiply to 8 and add to 6: that’s 4 and 2.
x 2 + 6 x + 8 = ( x + 4 ) ( x + 2 ) x^2 + 6x + 8 = (x + 4)(x + 2) x 2 + 6 x + 8 = ( x + 4 ) ( x + 2 )
Check (FOIL): ( x + 4 ) ( x + 2 ) = x 2 + 2 x + 4 x + 8 = x 2 + 6 x + 8 (x + 4)(x + 2) = x^2 + 2x + 4x + 8 = x^2 + 6x + 8 ( x + 4 ) ( x + 2 ) = x 2 + 2 x + 4 x + 8 = x 2 + 6 x + 8
3. Solve 5x − 7 = 3(x + 2)
5 x − 7 = 3 x + 6 5 x − 3 x = 6 + 7 2 x = 13 x = 13 2 = 6.5 \begin{aligned}
5x - 7 &= 3x + 6 \\[1em]
5x - 3x &= 6 + 7 \\[1em]
2x &= 13 \\[1em]
x &= \frac{13}{2} = 6.5
\end{aligned} 5 x − 7 5 x − 3 x 2 x x = 3 x + 6 = 6 + 7 = 13 = 2 13 = 6.5
4. Evaluate f(x) = 2x² − 3x + 1 when x = −1
f ( − 1 ) = 2 ( − 1 ) 2 − 3 ( − 1 ) + 1 = 2 ( 1 ) + 3 + 1 = 6 \begin{aligned}
f(-1) &= 2(-1)^2 - 3(-1) + 1 \\[1em]
&= 2(1) + 3 + 1 \\[1em]
&= 6
\end{aligned} f ( − 1 ) = 2 ( − 1 ) 2 − 3 ( − 1 ) + 1 = 2 ( 1 ) + 3 + 1 = 6
Algebra Basics skills solve practical problems:
Budget comparison : Two plans. Plan A: 45 + 0.12 x 45 + 0.12x 45 + 0.12 x 60 + 0.08 x 60 + 0.08x 60 + 0.08 x
45 + 0.12 x = 60 + 0.08 x 0.04 x = 15 x = 375 units \begin{aligned}
45 + 0.12x &= 60 + 0.08x \\[1em]
0.04x &= 15 \\[1em]
x &= 375 \text{ units}
\end{aligned} 45 + 0.12 x 0.04 x x = 60 + 0.08 x = 15 = 375 units
Area modeling : A room’s area is x 2 + 8 x + 15 x^2 + 8x + 15 x 2 + 8 x + 15
x 2 + 8 x + 15 = ( x + 3 ) ( x + 5 ) x^2 + 8x + 15 = (x + 3)(x + 5) x 2 + 8 x + 15 = ( x + 3 ) ( x + 5 )
So the room is ( x + 3 ) (x + 3) ( x + 3 ) ( x + 5 ) (x + 5) ( x + 5 )
Work rates : Job A pays 20 dollars/hour. Job B pays 15 dollars/hour plus a 100-dollar bonus. When does total pay match?
20 h = 15 h + 100 5 h = 100 h = 20 hours \begin{aligned}
20h &= 15h + 100 \\[1em]
5h &= 100 \\[1em]
h &= 20 \text{ hours}
\end{aligned} 20 h 5 h h = 15 h + 100 = 100 = 20 hours
These tools help compare costs, model areas and profits, and plan work or budgets.
You’ve covered a lot in Algebra Basics, from solving equations and systems to graphing lines and basic factoring. Review feels like a lot, but each skill connects to the next. Pick one problem type at a time, show your work, and check answers. You’re now equipped to handle many real-life math situations. Celebrate that progress.
Solve $4x + 5 = 2x + 17$. A. $x = 6$ B. $x = 4$ C. $x = 3$ D. $x = 12$
Factor $x^2 - 5x - 14$. A. $(x - 7)(x + 2)$ B. $(x + 7)(x - 2)$ C. $(x - 14)(x + 1)$ D. $(x + 14)(x - 1)$
A line has slope 3 and passes through $(1, 4)$. Which point-slope equation? A. $y - 4 = 3(x - 1)$ B. $y - 1 = 3(x - 4)$ C. $y = 3x + 4$ D. $y - 4 = 3x - 1$
Solve the system: $y = x + 2$ and $3x + y = 14$. A. $x = 3, y = 5$ B. $x = 4, y = 6$ C. $x = 2, y = 4$ D. $x = 5, y = 7$
What is the slope of the line through $(2, 5)$ and $(6, 13)$? A. $2$ B. $4$ C. $\frac{1}{2}$ D. $8$
Factor $x^2 + 7x + 12$. A. $(x + 3)(x + 4)$ B. $(x + 2)(x + 6)$ C. $(x + 1)(x + 12)$ D. $(x + 6)(x + 2)$
Solve $5x - 3 = 2x + 9$. A. $x = 4$ B. $x = 2$ C. $x = 6$ D. $x = 3$
What is the y-intercept of $y = -2x + 8$? A. $(0, 8)$ B. $(8, 0)$ C. $(0, -2)$ D. $(-2, 0)$
Simplify $3(2x - 4) + 5x$. A. $11x - 12$ B. $6x - 12$ C. $11x + 12$ D. $11x - 4$
Solve the system: $2x + y = 10$ and $x - y = 2$. A. $x = 4, y = 2$ B. $x = 3, y = 4$ C. $x = 5, y = 0$ D. $x = 6, y = -2$
Factor $x^2 - 9$. A. $(x - 3)(x + 3)$ B. $(x - 9)(x + 1)$ C. $(x - 3)^2$ D. $(x + 3)^2$
For $f(x) = 2x - 7$, what is $f(5)$?
Solve $-4x + 1 > 13$. A. $x < -3$ B. $x > -3$ C. $x < 3$ D. $x > 3$
What is the degree of $5x^3 - 2x + 8$?
A line passes through $(0, -3)$ with slope $2$. Write the equation. A. $y = 2x - 3$ B. $y = -3x + 2$ C. $y = 2x + 3$ D. $y = -2x - 3$
Add $(3x^2 + 2x - 1) + (x^2 - 5x + 4)$. A. $4x^2 - 3x + 3$ B. $4x^2 + 7x + 3$ C. $2x^2 - 3x + 3$ D. $4x^2 - 3x - 5$
Solve $\frac{x + 2}{3} = 5$. A. $x = 13$ B. $x = 17$ C. $x = 7$ D. $x = 9$
Two numbers add to 30 and differ by 8. What are they? A. $19$ and $11$ B. $20$ and $10$ C. $22$ and $8$ D. $18$ and $12$
A horizontal line has a slope of: A. $0$ B. Undefined C. $1$ D. $-1$
Factor completely: $3x^2 - 12$. A. $3(x - 2)(x + 2)$ B. $(3x - 6)(x + 2)$ C. $3(x^2 - 4)$ D. $(x - 2)(3x + 6)$
Retry Quiz
