test

Nov 11, 2025

测试作者

The problems analyzed from the "SAT Wall" are highly representative of the challenging, multi-concept questions often seen on the Digital SAT. The December test is expected to emphasize a mix of complex algebra, integrated geometry, and conceptual data analysis.

Here is a breakdown of the essential topics you must master to succeed.

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📈 Advanced Algebra and Functions

These questions often involve manipulating equations to meet a specific condition (like "no solution") or understanding the structure of polynomial expressions.

Concept	Problem Type	Core Skill
No Solution/Infinite Solutions	Linear equations with variables for coefficients (e.g., $m, n$ ).	Know the conditions for parallel lines: Slopes must be equal ( $m_{1} = m_{2}$ ), but y-intercepts must be different ( $b_{1} \neq = b_{2}$ ).
Polynomial Identity	Multiplying complex factors like $(a x + b) (c x^{2} + d x + e)$ and equating coefficients to a resulting polynomial.	Use the distributive property (FOIL/Box Method) and match the coefficients of corresponding powers of $x$ (e.g., $x^{3}$ , $x^{2}$ , $x$ ) to solve for unknown variables.
Sum of Solutions (Quadratics)	Solving equations that simplify to a quadratic, or using the sum/product rule.	For a quadratic $a x^{2} + b x + c = 0$ , the sum of solutions is $- b / a$ . Be able to factor and identify two solutions, even if one is easily found (e.g., a common factor).
Literal Equations	Rearranging scientific formulas (like the Law of Gravitation) to isolate a specific variable.	Treat variables as numbers and apply inverse operations (multiplication, division, squaring) to solve for the target variable.
Systems of Equations	Solving systems that involve linear and quadratic/non-linear equations.	Utilize the Desmos calculator to graph both equations and quickly find the points of intersection (solutions). This is the fastest method.

Integrated Geometry & Coordinate Geometry

The DSAT frequently tests geometry concepts by embedding them within algebra or data problems, requiring you to visualize a shape and its properties.

Concept	Problem Type	Core Skill
Circles (Standard Form)	Converting the general form of a circle's equation to the standard form $(x - h)^{2} + (y - k)^{2} = r^{2}$ .	Complete the Square for both $x$ and $y$ terms to find the center $(h, k)$ and the radius $r$ . Be prepared for coefficients greater than 1, requiring an initial division.
Tangent Lines & Slope	Finding the slope of a line tangent to a circle at a given point.	Understand that the radius to the point of tangency is perpendicular to the tangent line. Use the negative reciprocal of the radius slope for the tangent line's slope.
3D Shapes (Cones)	Finding volume using a right triangle formed by height, radius, and slant height.	Recognize that the height ( $H$ ), radius ( $R$ ), and slant height ( $S$ ) form a right triangle. Use $R^{2} + H^{2} = S^{2}$ to find the missing dimension before applying the volume formula.
Inscribed Shapes	Finding dimensions or area of a rectangle inscribed in a circle.	A rectangle's diagonal is the circle's diameter. Use this fact, often combined with the Pythagorean theorem or properties of 30-60-90 triangles, to solve.

Data Analysis, Probability, and Statistics

Be prepared for questions that test both conceptual understanding (like margin of error) and the ability to work with proportions and weighted averages.

Concept	Problem Type	Core Skill
Box Plots	Interpreting and comparing the distributions of two or more data sets.	Know that the line inside the box represents the median. Be able to compare quartiles and medians visually; the mean cannot be determined.
Weighted Averages	Finding the concentration/average of a mixture where components have unequal amounts (e.g., different carton numbers of different concentration milk).	Calculate the weighted total: $Total = \sum (Quantity \times Value) / \sum (Quantity)$ . Do not simply take the arithmetic mean.
Conditional Probability	Using proportions/percentages within a sample where subgroups are of unequal size (e.g., "Twice as many people prefer dark chocolate...").	Assign variables to the different group sizes (e.g., $N$ and $2 N$ ) and use them to weight the proportions before calculating the final probability.
Margin of Error (Conceptual)	Determining how changes in sample size affect the margin of error (MOE).	Inverse Relationship: A larger sample size results in a smaller margin of error (higher confidence).

🔥 New Challenge Predicted: Solving Intimidating Systems of Equations

The SAT Insight: Elimination is Your Friend 🤝

Instead of expanding the $6 (y - 9)$ terms and making the equations messier, notice that the $y$ terms in the two equations are opposites: $+ 6 (y - 9)$ and $- 6 (y - 9)$ . This is a massive hint to use the Elimination Method.

Step 1: Eliminate the Y-term by Adding the Equations

Add the left sides and the right sides: $(\frac{4}{x} + 6 (y - 9)) + (\frac{8}{x} - 6 (y - 9)) = 20 + 28$

The $y$ -terms cancel out perfectly: $\frac{4}{x} + \frac{8}{x} = 48$

Step 2: Solve for $x$

Combine the fractions on the left side: $\frac{4 + 8}{x} = 48$

\frac{12}{x} = 48

Now, solve for $x$ : $12 = 48 x$

x = \frac{12}{48} = \frac{1}{4}

Step 3: Calculate the Target Value ( $8 x$ )

The question asks for the value of $8 x$ , not just $x$ :

8 x = 8 \cdot (\frac{1}{4}) = 2

💡 Strategic Takeaway

Many of the "hard" problems (like those involving systems of equations or circle geometry) can be drastically simplified by effectively using the Desmos Calculator. Practice graphing non-standard equations and identifying key features, as this is a core time-saving strategy for the DSAT.

Good luck with your December exam preparation!