Step-by-Step Guidance for Systems of Linear Equations and Graphing Inequalities

Q1. Use the Substitution Method to solve the system:

$\begin{cases} 5x - 4y = 9 \\ x - 2y = -3 \end{cases}$

Background

Topic: Systems of Linear Equations (Substitution Method)

This question tests your ability to solve a system of two linear equations using the substitution method, which involves solving one equation for one variable and substituting that expression into the other equation.

Key Terms and Formulas

• System of equations: Two or more equations with the same variables.

• Substitution method: Solve one equation for one variable, then substitute into the other equation.

Step-by-Step Guidance

1. Solve the second equation for $x$ in terms of $y$:

   $x - 2y = -3$

   Isolate $x$:

   $x = -3 + 2y$

2. Substitute this expression for $x$ into the first equation:

   $5x - 4y = 9$ becomes $5(-3 + 2y) - 4y = 9$

3. Expand and simplify the equation to solve for $y$:

   $-15 + 10y - 4y = 9$

4. Combine like terms and isolate $y$.

Try solving on your own before revealing the answer!

Final Answer:

$y = 4$ and $x = 5$

Substituting $y$ back into the expression for $x$ gives the solution to the system.

Q2. Use the Addition Method to solve the system:

$\begin{cases} 3x + 2y = 4 \\ 2x + y = 1 \end{cases}$

Background

Topic: Systems of Linear Equations (Addition/Elimination Method)

This question tests your ability to solve a system of equations by adding or subtracting the equations to eliminate one variable.

Key Terms and Formulas

• Addition (Elimination) method: Multiply one or both equations (if needed) so that adding or subtracting the equations will eliminate one variable.

Step-by-Step Guidance

1. Align the equations for elimination. Notice that the $y$ terms are $2y$ and $y$.

2. Multiply the second equation by $-2$ to make the $y$ coefficients opposites:

   $-2(2x + y = 1) \implies -4x - 2y = -2$

3. Add the new equation to the first equation:

   $(3x + 2y) + (-4x - 2y) = 4 + (-2)$

4. Simplify to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer:

$x = 2$, $y = -1$

After finding $x$, substitute back into one of the original equations to find $y$.

Q3. Write the compound inequalities as a system:

$-3 \leq x < 2$ and $1 < y \leq 5$

Background

Topic: Systems of Inequalities

This question is about expressing compound inequalities as a system, which is a set of inequalities that must all be true at the same time.

Key Terms and Formulas

• Compound inequality: Two or more inequalities joined (here, by 'and').

• System of inequalities: A set of inequalities considered together.

Step-by-Step Guidance

1. Write each inequality separately:

   $-3 \leq x < 2$

   $1 < y \leq 5$

2. Express as a system using braces:

   $\begin{cases} -3 \leq x < 2 \\ 1 < y \leq 5 \end{cases}$

Try solving on your own before revealing the answer!

Final Answer:

The system is: $\begin{cases} -3 \leq x < 2 \\ 1 < y \leq 5 \end{cases}$

This expresses the solution set as a system of inequalities.

Q4. Graph $y > -\frac{2}{3}x$ without using a test point.

Background

Topic: Graphing Linear Inequalities

This question tests your understanding of how to graph a linear inequality and determine which region to shade, based on the inequality sign.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear function.

• Boundary line: The line $y = -\frac{2}{3}x$ (dashed for $>$ or $<$, solid for $\geq$ or $\leq$).

• Shading: For $y > mx + b$, shade above the line; for $y < mx + b$, shade below.

Step-by-Step Guidance

1. Graph the boundary line $y = -\frac{2}{3}x$ as a dashed line (since the inequality is strict, $>$).

2. Since the inequality is $y > -\frac{2}{3}x$, shade the region above the line.

3. No test point is needed because the rule is: for $y > mx + b$, always shade above the line.

Try solving on your own before revealing the answer!

Final Answer:

Draw a dashed line for $y = -\frac{2}{3}x$ and shade the region above the line.

This represents all points $(x, y)$ where $y$ is greater than $-\frac{2}{3}x$.

Q5. Graph $x^2 + y^2 \leq 9$.

Background

Topic: Graphing Circles and Inequalities

This question tests your ability to recognize and graph the region defined by a circle inequality.

Key Terms and Formulas

• Circle equation: $x^2 + y^2 = r^2$ is a circle centered at $(0,0)$ with radius $r$.

• Inequality $\leq$: Includes the interior and the boundary of the circle.

Step-by-Step Guidance

1. Recognize that $x^2 + y^2 = 9$ is a circle centered at $(0,0)$ with radius $3$.

2. Since the inequality is $\leq$, shade the region inside and on the circle.

3. Draw a solid circle with radius $3$ centered at the origin, and shade the interior.

Try solving on your own before revealing the answer!

Final Answer:

Graph a solid circle centered at $(0,0)$ with radius $3$, and shade the entire region inside the circle.

This represents all points $(x, y)$ such that $x^2 + y^2 \leq 9$.

Q6. Graph the solution set given by:

$\begin{cases} x - y < 1 \\ 2x + 3y \geq 12 \end{cases}$

Background

Topic: Graphing Systems of Linear Inequalities

This question tests your ability to graph the solution set for a system of two linear inequalities, which is the region where the shaded areas overlap.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear function.

• System of inequalities: The solution is the intersection (overlap) of the individual solution regions.

Step-by-Step Guidance

1. Graph the boundary line $x - y = 1$ as a dashed line (since the inequality is $<$).

2. Shade the region below the line $x - y = 1$ (where $x - y < 1$).

3. Graph the boundary line $2x + 3y = 12$ as a solid line (since the inequality is $\geq$).

4. Shade the region above or on the line $2x + 3y = 12$ (where $2x + 3y \geq 12$).

5. The solution set is the region where the two shaded areas overlap.

Try solving on your own before revealing the answer!

Final Answer:

The solution set is the region where the shaded areas for both inequalities overlap on the graph.

Be sure to use a dashed line for $x - y = 1$ and a solid line for $2x + 3y = 12$.