Solve the equations below for \(y\).
1) \( 2x+5y=10 \)
2) \( -3x+9y=12 \)
3) \( 12x−y=5 \)
4) \( 3x+4y−7=0 \)
1) \( 2x+5y=10 \)
2) \( -3x+9y=12 \)
3) \( 12x−y=5 \)
4) \( 3x+4y−7=0 \)
Given the student work below, identify the mistake and correct the work.
5) Problem: Solve for \(y\): \(2x+3y=12\)
Student work:
\( \begin{align*} 2x+3y&=12\\ 3y&=2x+12 \\ y&= \frac{2}{3}x+4 \end{align*} \)
6) Problem: Solve for \(y\): \(4x+2y=5\)
Student work:
\( \begin{align*} 4x+2y&=5 \\ 2y &=-4x+5 \\ y&= -2x+5 \end{align*} \)
5) Problem: Solve for \(y\): \(2x+3y=12\)
Student work:
\( \begin{align*} 2x+3y&=12\\ 3y&=2x+12 \\ y&= \frac{2}{3}x+4 \end{align*} \)
6) Problem: Solve for \(y\): \(4x+2y=5\)
Student work:
\( \begin{align*} 4x+2y&=5 \\ 2y &=-4x+5 \\ y&= -2x+5 \end{align*} \)
7) The equation for the perimeter of a rectangle is \( P=2l+2w \) where \(P\) is the perimeter, \(l\) is the length and \(w\) is the width. Solve the equation for the width.
8) The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is \( \rho \) (the lower case Greek letter rho). Mathematically, density is defined as mass divided by volume: \( \rho = \frac{m}{V} \). Solve the equation for \(V\).
9) The vector sum of the forces \(F\) on an object is equal to the mass m of that object multiplied by the acceleration vector \(a\) of the object: \(F=m⋅a\). Solve the equation for \(a\).
10) The standard form of a linear equation is \(Ax+By=C\). Solve the equation for \(x\).
11) The point-slope form for a linear equation is \(y−y_1=m(x−x_1)\). Solve the equation for \(x\).
12) The slope-intercept form for a linear equation is \(y=mx+b\). Solve the equation for \(x\).
13) The area equation for a trapezoid is \(A=\frac{1}{2}(b_1+b_2)h \). Where the area is \(A\), the \(b\) values are the lengths of the bases and \(h\) is the height. Solve the equation for one of the bases.
14) The Chemistry gas law is \( \frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2} \) where \(P\) is pressure, \(T\) is temperature and \(V\) is volume. Solve for \(V_2\).
15) Below is the following acid where \(M\) is the molarity and \(V\) is volume.
\( M_A V_A=M_B V_B\)
Solve for \(V_B\).
16) The Ideal Gas Law is \(PV=nRT\) where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is a constant and \(T\) is the temperature. Solve for \(R\).
17) Below is the area equation for a kite and the work to solve for one of the \(d\) values. Drag the red circles onto the blue circles and see if you can identify the correct justifications for each step.
8) The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is \( \rho \) (the lower case Greek letter rho). Mathematically, density is defined as mass divided by volume: \( \rho = \frac{m}{V} \). Solve the equation for \(V\).
9) The vector sum of the forces \(F\) on an object is equal to the mass m of that object multiplied by the acceleration vector \(a\) of the object: \(F=m⋅a\). Solve the equation for \(a\).
10) The standard form of a linear equation is \(Ax+By=C\). Solve the equation for \(x\).
11) The point-slope form for a linear equation is \(y−y_1=m(x−x_1)\). Solve the equation for \(x\).
12) The slope-intercept form for a linear equation is \(y=mx+b\). Solve the equation for \(x\).
13) The area equation for a trapezoid is \(A=\frac{1}{2}(b_1+b_2)h \). Where the area is \(A\), the \(b\) values are the lengths of the bases and \(h\) is the height. Solve the equation for one of the bases.
14) The Chemistry gas law is \( \frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2} \) where \(P\) is pressure, \(T\) is temperature and \(V\) is volume. Solve for \(V_2\).
15) Below is the following acid where \(M\) is the molarity and \(V\) is volume.
\( M_A V_A=M_B V_B\)
Solve for \(V_B\).
16) The Ideal Gas Law is \(PV=nRT\) where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is a constant and \(T\) is the temperature. Solve for \(R\).
17) Below is the area equation for a kite and the work to solve for one of the \(d\) values. Drag the red circles onto the blue circles and see if you can identify the correct justifications for each step.
18) A formula that links the two measures of temperature Fahrenheit (F) and Celsius (C) is \( \frac{9}{5} C = F-32 \).
Why would you need to solve the equation for either \(C\) or \(F\)? Use solving techniques and a real world example to defend your answer.
Review
19) Solve the equation: \(-7x - 5x = 19\)
20) Solve the equation: \(3x + 7 = 9 - 5x\)
21) Solve the equation: \(44 = -2(x - 11)\)
22) Solve the inequality: \(-4x + 5 \leq 9 + 4x\)
Solution Bank
Why would you need to solve the equation for either \(C\) or \(F\)? Use solving techniques and a real world example to defend your answer.
Review
19) Solve the equation: \(-7x - 5x = 19\)
20) Solve the equation: \(3x + 7 = 9 - 5x\)
21) Solve the equation: \(44 = -2(x - 11)\)
22) Solve the inequality: \(-4x + 5 \leq 9 + 4x\)
Solution Bank