The process of discerning if something is linear involves verifying whether a particular equation or function adheres to the characteristics of a linear system. Essentially, it should demonstrate additivity and scalar multiplication. If it meets these criteria, it's considered linear. Conversely, if it fails to meet these conditions, it's deemed nonlinear. This principle is a basic element in the realms of algebra and calculus.
| Topic | Problem | Solution |
|---|---|---|
| None | Solving a two-step equation with signed fractions… | Start with the equation \(-\frac{1}{2} y - \frac{6}{5} = -\frac{4}{3}\). |
| None | Consider the equation. \[ 2 x+5=8 x-4 \] What is … | Consider the equation \(2x + 5 = 8x - 4\). |
| None | olve for $u$. \[ 3 u=-\frac{9}{4} \] Simplify. yo… | The question is asking to solve for $u$ in the equation $3u = -\frac{9}{4}$. |
| None | An SUV is purchased new for $\$ 31,500$. Part: 0 … | An SUV is purchased new for $31,500. |
| None | $8 x-30=-2(x+5)$ | Distribute the -2 on the right side of the equation to both x and 5, we get \(8x - 30 = -2x - 10\). |
| None | Find the slope and the $y$-intercept of the line.… | The equation of the line is given in the slope-intercept form, which is \(y = mx + b\), where \(m\)… |
| None | 3. To surf the internet for 15 minutes at an airp… | Let's denote the cost for surfing the internet for \(x\) minutes as \(C(x)\). |
| None | Find the inverse of the matrix $A$ using minors, … | Find the determinant of the matrix \(A\): \[\text{det}(A) = -12\] |
| None | Measuring Temperature. The two most commonly used… | Given the Celsius temperature as -37 degrees, we need to convert this to Fahrenheit. The formula to… |
| None | Solve the equation, and check the solution. \[ 7 … | The given equation is \(7x = 6x + 4\). |
| None | Identify the slope and $y$-intercept, if they exi… | The given equation is in the form of a linear equation, \(y = mx + c\), where \(m\) is the slope an… |
| None | An SUV is purchased new for $\$ 31,500$. Part 1 o… | The linear function to represent the value of the vehicle t years after purchase, assuming that the… |
| None | Solve the following equation, and check the solut… | Start by simplifying the equation by moving all terms involving x to one side and the constants to … |
| None | Find the inverse of the matrix $A=\left[\begin{ar… | Given the matrix $A=\left[\begin{array}{cc}-1 & 1 \\ -2 & -8\end{array}\right]$, we are to find its… |
| None | Find the determinant of the matrix $\left[\begin{… | Find the determinant of the matrix \(\left[\begin{array}{cc}2 & 3 \\ -2 & 1\end{array}\right]\). |
| None | Solve the equation $-5 y+4=-11$. | Subtract 4 from both sides of the equation: \(-5y + 4 - 4 = -11 - 4\) |
| None | 29. The monthly rent for an $\mathbf{8 0 0}$ squa… | First, we need to find the slope (m), which is the change in rent divided by the change in square f… |
| None | A home purchased in 1996 for $\$ 181,262$ was app… | The problem is asking for a linear equation that describes the value of the home. The value of the … |
| None | $7 x+12=2(x+6)$ | Expand the right side of the equation to get \(7x + 12 = 2x + 12\) |
| None | Tight Knit is an online store that sells two diff… | Tight Knit is an online store that sells two different tiers of monthly subscription boxes with kni… |
| None | Solve for $z$. \[ -20=\frac{z}{10}-22 \] | Translate the equation into English: -20 = z/10 - 22 |
| None | Suppose that Marissa bought a new tablet for $\$ … | Suppose that Marissa bought a new tablet for $400. The resale value is expected to decrease by $40 … |
| None | $y=2 x+1$ | Given the equation \(y=2x+1\). |
| None | $2 x+3 y=$ | The question is incomplete, please provide a complete equation or a system of equations to solve. |
| None | 14. $-3 x+y=4$ | Rearrange the equation to solve for y in terms of x: \( -3x + y = 4 \) |
| None | Does \( (-40,45) \) make the equation \( y=-x \) … | Substitute \((x, y) = (-40, 45)\) into \(y = -x\). |
| None | Does \( (6,6) \) make the equation \( y=x \) true? | \(y=x\) |